This article is about the friar and patron saint. For other uses, see Francis of Assisi (disambiguation).
Saint Francis of Assisi, O.F.M.
Saint Francis of Assisi (Italian: San Francesco d'Assisi; born Giovanni di Pietro di Bernardone, but nicknamed Francesco (a tribute to France) by his father; 1181/1182Template:Spaced ndash October 3, 1226) was an Italian Catholic friar and preacher. He founded the men's Order of Friars Minor, the women's Order of St. Clare, and the Third Order of Saint Francis for men and women not able to live the lives of itinerant preachers, followed by the early members of the Order of Friars Minor, or the monastic lives of the Poor Clares. Francis is one of the most venerated religious figures in history.
Francis' father was Pietro di Bernardone, a prosperous silk merchant. Francis lived the high-spirited life typical of a wealthy young man, even fighting as a soldier for Assisi. While going off to war in 1204, Francis had a vision that directed him back to Assisi, where he lost his taste for his worldly life. On a pilgrimage to Rome, he joined the poor in begging at St. Peter's Basilica. The experience moved him to live in poverty. Francis returned home, began preaching on the streets, and soon gathered followers. His Order was authorized by Pope Innocent III in 1210. He then founded the Order of Poor Clares, which became an enclosed religious order for women, as well as the Order of Brothers and Sisters of Penance (commonly called the Third Order).
In 1219, he went to Egypt in an attempt to convert the Sultan to put an end to the conflict of the Crusades. By this point, the Franciscan Order had grown to such an extent that its primitive organizational structure was no longer sufficient. He returned to Italy to organize the Order. Once his community was authorized by the Pope, he withdrew increasingly from external affairs. In 1223, Francis arranged for the first Christmas nativity scene. In 1224, he received the stigmata, making him the first recorded person to bear the wounds of Christ's Passion. He died during the evening hours of October 3, 1226, while listening to a reading he had requested of Psalm 142(141).
On July 16, 1228, he was proclaimed a saint by Pope Gregory IX. He is known as the patron saint of animals and the environment, and is one of the two patron saints of Italy (with Catherine of Siena). It is customary for Catholic and Anglican churches to hold ceremonies blessing animals on his feast day of October 4. He is also known for his love of the Eucharist, his sorrow during the Stations of the Cross, and for the creation of the Christmas crèche or Nativity Scene.
Francis of Assisi was one of seven children born in late 1181 or early 1182 to Pietro and his wife Pica de Bourlemont, about whom little is known except that she was a noblewoman originally from Provence. Pietro was in France on business when Francis was born in Assisi, and Pica had him baptized as Giovanni. When his father returned to Assisi, he took to calling him Francesco ("the Frenchman"), possibly in honor of his commercial success and enthusiasm for all things French. Since the child was renamed in infancy, the change can hardly have had anything to do with his aptitude for learning French, as some have thought. As a youth, Francesco became a devotee of troubadours and was fascinated with all things Transalpine. Although many hagiographers remark about his bright clothing, rich friends, and love of pleasures, his displays of disillusionment toward the world that surrounded him came fairly early in his life, as is shown in the "story of the beggar." In this account, he was selling cloth and velvet in the marketplace on behalf of his father when a beggar came to him and asked for alms. At the conclusion of his business deal, Francis abandoned his wares and ran after the beggar. When he found him, Francis gave the man everything he had in his pockets. His friends quickly chided and mocked him for his act of charity. When he got home, his father scolded him in rage.
In 1201, he joined a military expedition against Perugia and was taken as a prisoner at Collestrada, spending a year as a captive. It is possible that his spiritual conversion was a gradual process rooted in this experience. Upon his return to Assisi in 1203, Francis returned to his carefree life. In 1204, a serious illness led him to a spiritual crisis. In 1205, Francis left for Apulia to enlist in the army of Walter III, Count of Brienne. A strange vision made him return to Assisi, deepening his ecclesiastical awakening.
According to the hagiographic legend, thereafter he began to avoid the sports and the feasts of his former companions. In response, they asked him laughingly whether he was thinking of marrying, to which he answered, "yes, a fairer bride than any of you have ever seen," meaning his "Lady Poverty". He spent much time in lonely places, asking God for enlightenment. By degrees he took to nursing lepers, the most repulsive victims in the lazar houses near Assisi. After a pilgrimage to Rome, where he joined the poor in begging at the doors of the churches, he said he had a mystical vision of Jesus Christ in the country chapel of San Damiano, just outside of Assisi, in which the Icon of Christ Crucified said to him, "Francis, Francis, go and repair My house which, as you can see, is falling into ruins." He took this to mean the ruined church in which he was presently praying, and so he sold some cloth from his father's store to assist the priest there for this purpose.
His father, Pietro, highly indignant, attempted to change his mind, first with threats and then with beatings. In the midst of legal proceedings before the Bishop of Assisi, Francis renounced his father and his patrimony, laying aside even the garments he had received from him in front of the public. For the next couple of months he lived as a beggar in the region of Assisi. Returning to the countryside around the town for two years, he embraced the life of a penitent, during which he restored several ruined chapels in the countryside around Assisi, among them the Porziuncola, the little chapel of St. Mary of the Angels just outside the town, which later became his favorite abode.
At the end of this period (on February 24, 1209, according to Jordan of Giano), Francis heard a sermon that changed his life forever. The sermon was about Matthew 10:9, in which Christ tells his followers they should go forth and proclaim that the Kingdom of Heaven was upon them, that they should take no money with them, nor even a walking stick or shoes for the road. Francis was inspired to devote himself to a life of poverty.
Clad in a rough garment, barefoot, and, after the Gospel precept, without staff or scrip, he began to preach repentance. He was soon joined by his first follower, a prominent fellow townsman, the jurist Bernardo di Quintavalle, who contributed all that he had to the work. Within a year Francis had eleven followers. Francis chose never to be ordained a priest, and the community lived as "lesser brothers," fratres minores in Latin. The brothers lived a simple life in the deserted lazar house of Rivo Torto near Assisi; but they spent much of their time wandering through the mountainous districts of Umbria, always cheerful and full of songs, yet making a deep impression upon their hearers by their earnest exhortations.
Francis' preaching to ordinary people was unusual since he had no license to do so. In 1209 he composed a simple rule for his followers ("friars"), the Regula primitiva or "Primitive Rule", which came from verses in the Bible.
The rule was "To follow the teachings of our Lord Jesus Christ and to walk in his footsteps." In 1209, Francis led his first eleven followers to Rome to seek permission from Pope Innocent III to found a new religious Order. Upon entry to Rome, the brothers encountered Bishop Guido of Assisi, who had in his company Giovanni di San Paolo, the Cardinal Bishop of Sabina. The Cardinal, who was the confessor of Pope Innocent III, was immediately sympathetic to Francis and agreed to represent Francis to the pope. Reluctantly, Pope Innocent agreed to meet with Francis and the brothers the next day. After several days, the pope agreed to admit the group informally, adding that when God increased the group in grace and number, they could return for an official admittance. The group was tonsured. This was important in part because it recognized Church authority and prevented his following from possible accusations of heresy, as had happened to the Waldensians decades earlier. Though Pope Innocent initially had his doubts, following a dream in which he saw Francis holding up the Basilica of St. John Lateran (the cathedral of Rome, thus the 'home church' of all Christendom), he decided to endorse Francis' Order. This occurred, according to tradition, on April 16, 1210, and constituted the official founding of the Franciscan Order. The group, then the "Lesser Brothers" (Order of Friars Minor also known as the Franciscan Order), preached on the streets and had no possessions. They were centered in the Porziuncola and preached first in Umbria, before expanding throughout Italy.
From then on, the new Order grew quickly with new vocations. Hearing Francis preaching in the church of San Rufino in Assisi in 1211, the young noblewoman Clare of Assisi became deeply touched by his message and realized her calling. Her cousin Rufino, the only male member of the family in their generation, was also attracted to the new Order (which he joined). On the night of Palm Sunday, March 28, 1212, Clare clandestinely left her family's palace. Francis received her at the Porziuncola and thereby established the Order of Poor Ladies, later called Poor Clares. This was an Order for women, and he gave Clare a religious habit, or garment, similar to his own, before lodging her and a few female companions in a nearby monastery of Benedictine nuns. Later he transferred them to San Damiano. There they were joined by many other women of Assisi. For those who could not leave their homes, he later formed the Third Order of Brothers and Sisters of Penance, a fraternity composed of either laity or clergy whose members neither withdrew from the world nor took religious vows. Instead, they observed the principles of Franciscan life in their daily lives. Before long, this Third Order grew beyond Italy.
Determined to bring the Gospel to all God's creatures, Francis sought on several occasions to take his message out of Italy. In the late spring of 1212, he set out for Jerusalem, but he was shipwrecked by a storm on the Dalmatian coast, forcing him to return to Italy. On May 8, 1213, he was given the use of the mountain of La Verna (Alverna) as a gift from Count Orlando di Chiusi, who described it as "eminently suitable for whoever wishes to do penance in a place remote from mankind." The mountain would become one of his favourite retreats for prayer.
Saint Francis Abandons His Father. Francis of Assisi breaking off his relationship with his father and renouncing his patrimony, laying aside publicly even the garments he had received from him.
In the same year, Francis sailed for Morocco, but this time an illness forced him to break off his journey in Spain. Back in Assisi, several noblemen (among them Tommaso da Celano, who would later write the biography of St. Francis) and some well-educated men joined his Order. In 1215, Francis went again to Rome for the Fourth Lateran Council. During this time, he probably met a canon, Dominic de Guzman (later to be Saint Dominic, the founder of the Friars Preachers, another Catholic religious order). In 1217, he offered to go to France. Cardinal Ugolino of Segni (the future Pope Gregory IX), an early and important supporter of Francis, advised him against this and said that he was still needed in Italy.
In 1219, accompanied by another friar and hoping to convert the Sultan of Egypt or win martyrdom in the attempt, Francis went to Egypt where a Crusader army had been encamped for over a year besieging the walled city of Damietta two miles (Script error: No such module "convert".) upstream from the mouth of one of the main channels of the Nile. The Sultan, al-Kamil, a nephew of Saladin, had succeeded his father as Sultan of Egypt in 1218 and was encamped upstream of Damietta, unable to relieve it. A bloody and futile attack on the city was launched by the Christians on August 29, 1219, following which both sides agreed to a ceasefire which lasted four weeks. It was most probably during this interlude that Francis and his companion crossed the Saracen lines and were brought before the Sultan, remaining in his camp for a few days. The visit is reported in contemporary Crusader sources and in the earliest biographies of Francis, but they give no information about what transpired during the encounter beyond noting that the Sultan received Francis graciously and that Francis preached to the Saracens without effect, returning unharmed to the Crusader camp. No contemporary Arab source mentions the visit. One detail, added by Bonaventure in the official life of Francis (written forty years after the event), has Francis offering to challenge the Sultan's "priests" to trial-by-fire in order to prove the veracity of the Christian Gospel.
Such an incident is alluded to in a scene in the late 13th-century fresco cycle, attributed to Giotto, in the upper basilica at Assisi (see accompanying illustration). It has been suggested that the winged figures atop the columns piercing the roof of the building on the left of the scene are not idols (as Erwin Panofsky had proposed) but are part of the secular iconography of the sultan, affirming his worldly power which, as the scene demonstrates, is limited even as regards his own "priests" who shun the challenge. Although Bonaventure asserts that the sultan refused to permit the challenge, subsequent biographies went further, claiming that a fire was actually kindled which Francis unhesitatingly entered without suffering burns. The scene in the fresco adopts a position mid-way between the two extremes.
According to some late sources, the Sultan gave Francis permission to visit the sacred places in the Holy Land and even to preach there. All that can safely be asserted is that Francis and his companion left the Crusader camp for Acre, from where they embarked for Italy in the latter half of 1220. Drawing on a 1267 sermon by Bonaventure, later sources report that the Sultan secretly converted or accepted a death-bed baptism as a result of the encounter with Francis. The Franciscan Order has been present in the Holy Land almost uninterruptedly since 1217 when Brother Elias arrived at Acre. It received concessions from the Mameluke Sultan in 1333 with regard to certain Holy Places in Jerusalem and Bethlehem, and (so far as concerns the Catholic Church) jurisdictional privileges from Pope Clement VI in 1342.
Oldest known portrait in existence of the saint, dating back to St. Francis' retreat to Subiaco (1223–1224): depicted without the stigmata.
By this time, the growing Order of friars was divided into provinces and groups were sent to France, Germany, Hungary, and Spain and to the East. Upon receiving a report of the martyrdom of five brothers in Morocco, Francis returned to Italy via Venice. Cardinal Ugolino di Conti was then nominated by the Pope as the protector of the Order. Another reason for Francis' return to Italy was that the friars in Italy were causing problems. The Franciscan Order had grown at an unprecedented rate compared to prior religious orders, but its organizational sophistication had not kept up with this growth and had little more to govern it than Francis' example and simple rule. To address this problem, Francis prepared a new and more detailed Rule, the "First Rule" or "Rule Without a Papal Bull" (Regula prima, Regula non bullata), which again asserted devotion to poverty and the apostolic life. However, it also introduced greater institutional structure though this was never officially endorsed by the pope.
On September 29, 1220, Francis handed over the governance of the Order to Brother Peter Catani at the Porziuncola, but Brother Peter died only five months later, on March 10, 1221, and was buried there. When numerous miracles were attributed to the deceased brother, people started to flock to the Porziuncola, disturbing the daily life of the Franciscans. Francis then prayed, asking Peter to stop the miracles and to obey in death as he had obeyed during his life.
The reports of miracles ceased. Brother Peter was succeeded by Brother Elias as Vicar of Francis. Two years later, Francis modified the "First Rule", creating the "Second Rule" or "Rule With a Bull", which was approved by Pope Honorius III on November 29, 1223. As the official Rule of the Order, it called on the friars "to observe the Holy Gospel of our Lord Jesus Christ, living in obedience without anything of our own and in chastity". In addition, it set regulations for discipline, preaching, and entry into the Order. Once the Rule was endorsed by the Pope, Francis withdrew increasingly from external affairs. During 1221 and 1222, Francis crossed Italy, first as far south as Catania in Sicily and afterwards as far north as Bologna.
While he was praying on the mountain of Verna, during a forty-day fast in preparation for Michaelmas (September 29), Francis is said to have had a vision on or about September 14, 1224, the Feast of the Exaltation of the Cross, as a result of which he received the stigmata. Brother Leo, who had been with Francis at the time, left a clear and simple account of the event, the first definite account of the phenomenon of stigmata. "Suddenly he saw a vision of a seraph, a six-winged angel on a cross. This angel gave him the gift of the five wounds of Christ." Suffering from these stigmata and from trachoma, Francis received care in several cities (Siena, Cortona, Nocera) to no avail. In the end, he was brought back to a hut next to the Porziuncola. Here, in the place where it all began, feeling the end approaching, he spent the last days of his life dictating his spiritual Testament. He died on the evening of Saturday, October 3, 1226, singing Psalm 142 (141), "Voce mea ad Dominum". On July 16, 1228, he was pronounced a saint by Pope Gregory IX (the former cardinal Ugolino di Conti, friend of St. Francis and Cardinal Protector of the Order). The next day, the Pope laid the foundation stone for the Basilica of Saint Francis in Assisi. Francis was buried on May 25, 1230, under the Lower Basilica, but his tomb was soon hidden on orders of Brother Elias to protect it from Saracen invaders. His exact burial place remained unknown until it was re-discovered in 1818. Pasquale Belli then constructed for the remains a crypt in neo-classical style in the Lower Basilica. It was refashioned between 1927 and 1930 into its present form by Ugo Tarchi, stripping the wall of its marble decorations. In 1978, the remains of St. Francis were examined and confirmed by a commission of scholars appointed by Pope Paul VI, and put into a glass urn in the ancient stone tomb.
It has been argued that no one else in history was as dedicated as Francis to imitate the life, and carry out the work of Christ, in Christ's own way. This is important in understanding Francis' character and his affinity for the Eucharist and respect for the priests who carried out the sacrament.
He and his followers celebrated and even venerated poverty. Poverty was so central to his character that in his last written work, the Testament, he said that absolute personal and corporate poverty was the essential lifestyle for the members of his Order.
Francis' visit to Egypt and attempted rapprochement with the Muslim world had far-reaching consequences, long past his own death, since after the fall of the Crusader Kingdom, it would be the Franciscans, of all Catholics, who would be allowed to stay on in the Holy Land and be recognized as "Custodians of the Holy Land" on behalf of the Catholic Church.
At Greccio near Assisi, around 1220, Francis celebrated Christmas by setting up the first known presepio or crèche (Nativity scene). His nativity imagery reflected the scene in traditional paintings. He used real animals to create a living scene so that the worshipers could contemplate the birth of the child Jesus in a direct way, making use of the senses, especially sight. Thomas of Celano, a biographer of Francis and Saint Bonaventure both, tell how he used only a straw-filled manger (feeding trough) set between a real ox and donkey. According to Thomas, it was beautiful in its simplicity, with the manger acting as the altar for the Christmas Mass.
Francis preached the teaching of the Catholic Church, that the world was created good and beautiful by God but suffers a need for redemption because of the primordial sin of man. He preached to man and beast the universal ability and duty of all creatures to praise God (a common theme in the Psalms) and the duty of men to protect and enjoy nature as both the stewards of God's creation and as creatures ourselves. On November 29, 1979, Pope John Paul II declared St. Francis the Patron Saint of Ecology. Many of the stories that surround the life of St. Francis say that he had a great love for animals and the environment.
Perhaps the most famous incident that illustrates the Saint's humility towards nature is recounted in the "Fioretti" ("Little Flowers"), a collection of legends and folklore that sprang up after the Saint's death. It is said that, one day, while Francis was travelling with some companions, they happened upon a place in the road where birds filled the trees on either side. Francis told his companions to "wait for me while I go to preach to my sisters the birds." The birds surrounded him, intrigued by the power of his voice, and not one of them flew away. He is often portrayed with a bird, typically in his hand.
Another legend from the Fioretti tells that in the city of Gubbio, where Francis lived for some time, was a wolf "terrifying and ferocious, who devoured men as well as animals." Francis had compassion upon the townsfolk, and so he went up into the hills to find the wolf. Soon, fear of the animal had caused all his companions to flee, though the saint pressed on. When he found the wolf, he made the sign of the cross and commanded the wolf to come to him and hurt no one. Miraculously the wolf closed his jaws and lay down at the feet of St. Francis.
"Brother Wolf, you do much harm in these parts and you have done great evil," said Francis. "All these people accuse you and curse you...But brother wolf, I would like to make peace between you and the people." Then Francis led the wolf into the town, and surrounded by startled citizens made a pact between them and the wolf. Because the wolf had "done evil out of hunger, the townsfolk were to feed the wolf regularly. In return, the wolf would no longer prey upon them or their flocks. In this manner Gubbio was freed from the menace of the predator. Francis even made a pact on behalf of the town dogs, that they would not bother the wolf again. Finally, to show the townspeople that they would not be harmed, Francis blessed the wolf.
Then during the World Environment Day 1982, John Paul II said that St. Francis' love and care for creation was a challenge for contemporary Catholics and a reminder "not to behave like dissident predators where nature is concerned, but to assume responsibility for it, taking all care so that everything stays healthy and integrated, so as to offer a welcoming and friendly environment even to those who succeed us." The same Pope wrote on the occasion of the World Day of Peace, January 1, 1990, the saint of Assisi "offers Christians an example of genuine and deep respect for the integrity of creation..." He went on to make the point that: "As a friend of the poor who was loved by God's creatures, Saint Francis invited all of creation – animals, plants, natural forces, even Brother Sun and Sister Moon – to give honor and praise to the Lord. The poor man of Assisi gives us striking witness that when we are at peace with God we are better able to devote ourselves to building up that peace with all creation which is inseparable from peace among all peoples."
Pope John Paul II concluded that section of the document with these words, "It is my hope that the inspiration of Saint Francis will help us to keep ever alive a sense of 'fraternity' with all those good and beautiful things which Almighty God has created."
Francis of Assisi Francisco de Zurbarán. Saint Francis' feast day is observed on October 4. The Evangelical Church in Germany, however, commemorates St. Francis' feast day on his death day, October 3.
Saint Francis' feast day is observed on October 4. A secondary feast in honor of the stigmata received by St. Francis, celebrated on September 17, was inserted in the General Roman Calendar in 1585 (later than the Tridentine Calendar) and suppressed in 1604, but was restored in 1615. In the New Roman Missal of 1969, it was removed again from the General Calendar, as something of a duplication of the main feast on October 4, and left to the calendars of certain localities and of the Franciscan Order. Wherever the traditional Roman Missal is used, however, the feast of the Stigmata remains in the General Calendar.
On June 18, 1939, Pope Pius XII named Francis a joint Patron Saint of Italy along with Saint Catherine of Siena with the apostolic letter "Licet Commissa". Pope Pius also mentioned the two saints in the laudative discourse he pronounced on May 5, 1949, in the Church of Santa Maria sopra Minerva.
St. Francis is honored in the Church of England, the Anglican Church of Canada, the Episcopal Church USA, the Old Catholic Churches, the Evangelical Lutheran Church in America, and other churches and religious communities on October 4. The Evangelical Church in Germany, however, commemorates St. Francis' feast day on his death day, October 3.
On 13 March 2013, upon his election as Pope, Archbishop and Cardinal Jorge Mario Bergoglio of Argentina chose Francis as his papal name in honor of Saint Francis of Assisi, becoming Pope Francis.
Canticum Fratris Solis or Laudes Creaturarum; Canticle of the Sun.
For a complete list, see The Franciscan Experience.
Saint Francis is considered the first Italian poet by literary critics. He believed commoners should be able to pray to God in their own language, and he wrote often in the dialect of Umbria instead of Latin. His writings are considered to have great literary and religious value.
The anonymous 20th-century prayer "Make Me an Instrument of Your Peace" is widely but erroneously attributed to St. Francis.
The Franciscan Order promoted devotion to the life of Saint Francis from his canonization onwards, and commissioned large numbers of works for Franciscan churches, either showing St Francis with sacred figures, or episodes from his life. There are large early fresco cycles in the Basilica of San Francesco d'Assisi, parts of which are shown above.
Statue of St. Francis in front of the Catholic church of Chania.
Sant Francesc (Saint Francis, 1895), a book of forty-three Saint Francis poems by Catalan poet-priest Jacint Verdaguer, three of which are included in English translation in Selected Poems of Jacint Verdaguer: A Bilingual Edition, edited and translated by Ronald Puppo, with an introduction by Ramon Pinyol i Torrents (University of Chicago, 2007). The three poems are "The Turtledoves", "Preaching to Birds" and "The Pilgrim".
Saint Francis of Assisi a Doubleday Image Book translated by T. O'Conor Sloane, Ph.D., LL.D. in 1955 from the Danish original researched and written by Johannes Jorgensen and published in 1912 by Longmans, Green and Company, Inc.
John Tolan, St. Francis and the Sultan: The Curious History of a Christian-Muslim Encounter. Oxford: Oxford University Press, 2009.
Vita di un uomo: Francesco d'Assisi (1995) a book by Chiara Frugoni, preface by Jacques Le Goff, Torino: Einaudi.
Francis, Brother of the Universe (1982), a 48-page comic book by Marvel Comics on the life of Saint Francis of Assisi written by Father Roy Gasnik O.F.M. and Mary Jo Duffy, artwork by John Buscema and Marie Severin, lettering by Jim Novak and edited by Jim Shooter.
"Universal Man", an illumination from a 13th-century copy of Hildegard von Bingen's Liber Divinorum Operum ("Book of Divine Works", c. 1165).
In Rubén Darío's poem Los Motivos Del Lobo (The Reasons Of The Wolf) St. Francis tames a terrible wolf only to discover that the human heart harbors darker desires than those of the beast.
In Fyodor Dostoyevsky's The Brothers Karamazov, Ivan Karamazov invokes the name of 'Pater Seraphicus,' an epithet applied to St. Francis, to describe Alyosha's spiritual guide Zosima. The reference is found in Goethe's "Faust," Part 2, Act 5, lines 11918–25.
Rich Mullins co-wrote Canticle of the Plains, a musical, with Mitch McVicker. Released in 1997, it was based on the life of St. Francis of Assisi, but told as a western story.
Bernard Malamud's novel The Assistant (1957) features a protagonist, Frank Alpine, who exemplifies the life of St. Francis in mid-20th-century Brooklyn, New York City.
Canticle of the Sun is a prayer by St. Francis.
Prayer of Saint Francis is not a prayer by St. Francis.
^ On the day of his election, the Vatican clarified that his official papal name was "Francis", not "Francis I". A Vatican spokesman said that the name would become Francis I if and when there is a Francis II.
^ a b c d e f g h i j k l m n o p q r Brady, Ignatius Charles. "Saint Francis of Assisi." Encyclopædia Britannica Online.
^ a b c d e f g h i j 12px Paschal Robinson (1913). "St. Francis of Assisi". Catholic Encyclopedia. New York: Robert Appleton Company.
^ a b c d e f Cross, F. L., ed. (2005). "Francis of Assisi". The Oxford dictionary of the Christian church. New York: Oxford University Press. ISBN 0199566712.
^ Tolan, John (2009). St. Francis and the Sultan: The Curious History of a Christian-Muslim Encounter. Oxford: Oxford University Press. ISBN 9780199239726.
^ Cross, F. L., ed. (2005). "Stigmatization". The Oxford dictionary of the Christian church. New York: Oxford University Press. ISBN 0199566712.
^ a b "Blessing All Creatures, Great and Small". Duke Magazine. November 2006. Retrieved 2007-07-30.
^ "St. Francis of Assisi – Franciscan Friars of the Renewal". Franciscanfriars.com. Retrieved 24 October 2012.
^ under subheading "The Crib (Creche) or Nativity Scene"
^ a b Englebert, Omer (1951). The Lives of the Saints. New York: Barnes & Noble. p. 529. ISBN 978-1-56619-516-4.
^ Robinson, P. (2009). St. Francis of Assisi. In The Catholic Encyclopedia. New York: Robert Appleton Company. Retrieved 2011-10-17 from New Advent.
^ a b Chesterton, Gilbert Keith (1924). "St. Francis of Assisi" (14 ed.). Garden City, New York: Image Books. p. 158.
^ Bonaventure; Cardinal Manning (1867). The Life of St. Francis of Assisi (from the Legenda Sancti Francisci) (1988 ed.). Rockford, Illinois: TAN Books & Publishers. p. 190. ISBN 978-0-89555-343-0.
^ Fioretti quoted in: St. Francis, The Little Flowers, Legends, and Lauds, trans. N. Wydenbruck, ed. Otto Karrer (London: Sheed and Ward, 1979) 244.
^ Steven Runciman, History of the Crusades, vol. 3: The Kingdom of Acre and the Later Crusades, Cambridge University Press (1951, paperback 1987), pp. 151–161.
^ e.g., Jacques de Vitry, Letter 6 of February or March 1220 and Historia orientalis (c. 1223–1225) cap. XXII; Tommaso da Celano, Vita prima (1228), §57: the relevant passages are quoted in an English translation in Tolan, pp. 19f. and 54 respectively.
^ e.g., Chesterton, Saint Francis, Hodder & Stoughton (1924) chapter 8. Tolan (p.126) discusses the incident as recounted by Bonaventure which does not extend to a fire actually being lit.
^ Péter Bokody, "Idolatry or Power: St. Francis in Front of the Sultan," in: Promoting the Saints: Cults and Their Contexts from Late Antiquity until the Early Modern Period, ed. Ottó Gecser and others (Budapest: CEU Press, 2010), 69-81, esp. at pp. 74 and 76-78. The views of Panofsky (idols: see Renaissance and Renascences in Western Art, New York 1972, p.148, n.3) and Tolan (undecided: p.143) are cited at p.73.
^ Ugolino Brunforte (Brother Ugolino). The Little Flowers of St. Francis of Assisi. Calvin College: CCEL. ISBN 1-61025212-8. ISBN 978-1-61025212-6. Quote.
^ Pope John Paul II (November 29, 1979). "Inter Sanctos (Apostolic Letter AAS 71)" (PDF). Retrieved August 7, 2014.
^ Pope John Paul II (December 8, 1989). "World Day of Peace 1990". Retrieved October 24, 2012.
^ Pope Francis (March 16, 2013). "Audience to Representatives of the Communications Media". Retrieved August 9, 2014.
^ "Pope Francis explains decision to take St Francis of Assisi's name". London: The Guardian. 16 March 2013. Archived from the original on 16 March 2013.
^ a b "New Pope Fra[n]cis visits St. Mary Major, collects suitcases and pays bill at hotel". NEWS.VA. 14 March 2013. Archived from the original on 16 March 2013.
^ Michael Martinez, CNN Vatican analyst: Pope Francis' name choice 'precedent shattering', CNN (13 March 2013). Retrieved 13 March 2013.
^ "Pope Francis wants 'poor Church for the poor'". BBC News (BBC). 16 March 2013. Retrieved 16 March 2013.
^ Alpert, Emily (13 March 2013). "Vatican: It's Pope Francis, not Pope Francis I". Los Angeles Times. Archived from the original on 16 March 2013.
^ Brand, Peter; Pertile, Lino, eds. (1999). "2 - Poetry. Francis of Assisi (pp. 5ff.)". The Cambridge History of Italian Literature. Cambridge University Press. ISBN 0-52166622-8. ISBN 978-0-52166622-0.
^ Chesterton, G.K. (1987). St. Francis. Image. pp. 160 p. ISBN 0-385-02900-4.
^ Renoux, Christian (2001). La prière pour la paix attribuée à saint François: une énigme à résoudre. Paris: Editions franciscaines. ISBN 2-85020-096-4.
^ Renoux, Christian. "The Origin of the Peace Prayer of St. Francis". Retrieved August 9, 2014.
Englebert, Omer (1951). The Lives of the Saints. New York: Barnes & Noble.
Robinson, Paschal (1913). "St. Francis of Assisi". Catholic Encyclopedia. New York: Robert Appleton Company. 12px "St._Francis_of_Assisi". Catholic Encyclopedia. New York: Robert Appleton Company. 1913.
Tolan, John (2009). Saint Francis and the Sultan. Oxford: Oxford University Press.
Friar Elias, Epistola Encyclica de Transitu Sancti Francisci, 1226.
Pope Gregory IX, Bulla "Mira circa nos" for the canonization of St. Francis, July 19, 1228.
Friar Tommaso da Celano: Vita Prima Sancti Francisci, 1228; Vita Secunda Sancti Francisci, 1246–1247; Tractatus de Miraculis Sancti Francisci, 1252–1253.
Friar Julian of Speyer, Vita Sancti Francisci, 1232–1239.
St. Bonaventure of Bagnoregio, Legenda Maior Sancti Francisci, 1260–1263.
Ugolino da Montegiorgio, Actus Beati Francisci et sociorum eius, 1327–1342.
Fioretti di San Francesco, the "Little Flowers of St. Francis", end of the 14th century: an anonymous Italian version of the Actus; the most popular of the sources, but very late and therefore not the best authority by any means.
Acocella, Joan, "Rich Man, Poor Man: The Radical Visions of St. Francis", The Critics: Books, The New Yorker 88, no. 43 (January 14, 2013): 72-77 (accessed January 23, 2015).
St. Francis of Assisi at the Christian Iconography web site.
"All Creatures of Our God and King"
"Légende No. 1: St François d'Assise"


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We need to start taking our passwords seriously, come on
A raft of recent research has once again revealed that many of us are still absolutely terrible when it comes to creating strong password. This includes a new report from password manager Nordpass, which examined a database that totals over 3TB of compromised passwords and spans users from 30 countries in order to reveal the top 200 passwords, ranking entries by how many instances were found, how easy they were to crack, in addition to popularity by country and, where possible, gender. Perhaps unsurprisingly, Nordpass found that "password" remains the…
Beware – Black Friday online shopping scams are here now
Although Black Friday isn't due for another couple of days, cybercriminals aren't holding back from trying to take advantage of the upcoming shopping spree. Cybersecurity researchers from Bitdefender have carried out a wide range of recent analysis, and found numerous scams trying to take advantage of gullible shoppers. The most popular method seems to be a phishing campaign luring people with huge discounts on designer bags and sunglasses, and then stealing their credit card information, when victims submit this information on specially-crafted landing pages. Fake surveys, fake sunglasses Besides fake…
Garmin DriveCam 76 review
Two-minute review You'd be forgiven for thinking that dedicated sat navs were a thing of the past, usurped first by integrated systems from car manufacturers, then by smartphones. This may be true for many drivers, but for those who want a navigation system that promises to outshine both car and phone, a dedicated unit can still be the way to go. Especially if you are a habitual road-tripper keen for international adventure, and would rather keep your smartphone hidden safely away from the windshield. The Garmin DriveCam 76 fits this…
Vantrue N2 Pro review
Two-minute review You're not alone in thinking the Vantrue N2 Pro is a curious-looking dash cam. The angular body with buttons, lenses and design details seemingly scattered across every surface, is unlikely to win many awards on aesthetics alone. But not many of the world's best dash cams are design pieces, so we're willing to mostly skip over the design of the N2 Pro and instead focus on its features, functions and performance. Although not attractive, the Vantrue is reasonably compact considering it houses a pair of cameras, plus a…
Azure will now protect SMBs from DDoS attacks
Microsoft have announced that a new SKU for its cloud computing platform Azure's DDoS Protection offering, which it says is specifically designed with SMBs in mind, is now available in public preview. Dubbed "IP Protection" the latest SKU is built to defend against L3/L4 DDoS attacks via "always-on monitoring and adaptive tuning". According to Microsoft, IP Protection integrates with other Azure services such as Azure Firewall Manager, Microsoft Sentinel, and Microsoft Defender for Cloud to provide real-time alerts and metrics to enable protection for a single public IP. Azure's new…
No, you're not having an identity crisis – there could be strangers in your iCloud photos
iCloud for Windows app users are reporting some fairly bizarre glitches with photo and video content when accessing the platform on a PC. Numerous complaints on the MacRumours forum detail users' purported experiences of photos and even videos of strangers appearing in their library on the iCloud app for Windows. Also, videos apparently shot with the iPhone 13 Pro and 14 Pro are glitched when synced with the platform, with blank screens and scan lines being all that remain. The problem appears isolated to Windows machines, and specifically with the…
This nasty Google Chrome extension is after your crypto and your passwords
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redpajama_set_name: RedPajamaCommonCrawl


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Home > Retail Products and Services > Specialty Store
Adam & Eve Franchise Review
How much does an Adam & Eve franchise cost?
Adam & Eve has a franchise fee of up to $30,000, with a total initial investment range of $171,050 to $350,700.
Total Investment: $171,050 to $350,700
Working Capital: $5,000 to $25,000
How much does an Adam & Eve franchise make?
As of the 2016 Franchise Disclosure Document, there are 64 franchised Adam & Eve locations in the USA.
Are there any Adam & Eve franchise opportunities near me?
Based on 2016 FDD data, Adam & Eve has franchise locations in 0 states. The largest region is the West with 0 franchise locations.
Adam & Eve is a Franchise Grade Made The Grade™ franchise, which means it's ranked highly by our analysts!
Made the
Adam & Eve Franchise Stats
Yes, they have an initial training program of 63 hours. This includes 45 hours of classroom training and 18 hours of on-the-job training. In their FDD they outline who's in charge of the training. Unlock to find out more.
Adam & Eve offers territory protections. For an explanation of territorial rights, unlock this franchise for important details.
They began franchising in 2004. The average Specialty Store franchise began franchising in 2012.
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SAN FRANCISCO, Oct. 12, 2017 /PRNewswire/ -- Omnyway, the leading augmented digital commerce platform, announced today that it has closed on $12.75 million in Series A funding to help Omnyway grow its operations, launch new services on its current contextual commerce platform, and expand its customer base in the U.S. and worldwide. The round was led by Nyca, CommerzVentures, the venture arm of Commerzbank Group of Germany, and CEFIF, the CreditEase Fintech Investment Fund, a part of CreditEase of China, with participation from Commerce Ventures, Industry Ventures and others.
Omnyway empowers retailers and brands to build an augmented digital commerce ecosystem that brings the power of the virtual world to the shoppers experience in the physical store, using the mobile phone as the bridge for all aspects of their buying journey. The Omnyway platform creates a dynamic digital channel between the retailer and the shopper across all channels of interaction—in store, online, in-app, virtual aisle and dynamic media, using any smart mobile device. Since the company was founded in 2014, it has established itself as a significant player in retail commerce and continues to see strong interest in its platform. This funding is a milestone that acknowledges Omnyway's momentum to-date and that will set the company up for sustained growth over the long-term.
"To keep up in this fast-moving, mobile-first (and soon, mobile-only) world, retailers need tools that speak to how people want to shop and engage today," said Omnyway CEO, Ashok Narasimhan. "This investment puts us in a great position to build on growth and continue innovating on today's fundamentally different retail experience."
In addition to its digital platform, Omnyway has seen strong interest in its newly launched Zapbuy service for retailers, brands and financial institutions. Zapbuy allows consumers to instantly buy items directly from ads with just one click of the ad. With digital advertising spend projected to be a massive $44.8 billion in 2017, an overwhelming majority of marketers are frustrated with their ROI. In most cases, the path that leads from a consumer view an ad to the actual purchase is inconvenient and full of friction. Omnyway solves this problem with Zapbuy. Shoppers can complete a purchase in a single step, enabling retailers and brands to deliver a seamless buying experience and significantly increase conversion rates, while allowing a customer to stay within the context (e.g., social media) where the customer encountered the ads.
"Every single retailer is seeking ways to make customer experiences seamless, customized, and integrated in-store and across channels," said Hans Morris, Managing Partner of Nyca. "This has turned out to be expensive and difficult, and the flexibility, security, scalability—and speed to market -- of the Omnyway platform is a compelling solution."
"CommerzVentures supports ambitious entrepreneurs that are driven by the vision of redefining the financial services industry," said Stefan Tirtey, Managing Director of CommerzVentures. "In Omnyway, we see a company thinking creatively about the future of retail in the new world. We look forward to supporting the team as it scales up."
"Mobile devices are driving a radical transformation in the payments space," said Anju Patwardhan, Managing Director of CEFIF. "We are always looking to partner with entrepreneurs and companies that are helping to redefine FinTech and are thrilled to be a part of Omnyway's journey as it expands globally and introduces new payment innovations"
Nyca Partners is a venture capital firm exclusively focused on applying innovation in financial services into the global financial system. Our rich experience and deep connections in both finance and technology give us a unique perspective and ability to help entrepreneurs transform payments, credit models, digital advice, and financial infrastructure. We strive to form truly collaborative partnerships, offering investment capital and expert advice. The investment team is comprised of Hans Morris, Jeff Reitman, and David Sica; with Ravi Mohan as the COO. The Investment Partners are Osama Bedier, Brian Finn, Tom Miglis, and Charlie Songhurst.
Founded in October 2014, CommerzVentures operates as the corporate venture capital fund of Commerzbank Group. We back early- and growth-stage companies in the financial services and insurance sectors. We aspire to invest in the very best players in a given market segment, always pursuing our main objective of generating financial return.
Founded in February 2016, CreditEase Fintech Investment Fund is a venture fund investing in growth-stage fintech companies in China and the global markets. CreditEase Fintech Investment Fund has an equivalent of $1 billion in total committed capital. The fund has formed strategic partnerships with global leading venture capital investors to discover opportunities in five sub-segments within the domain of Fintech: Lending, Payment, Personal Finance/Wealth management, Enterprise Solutions and Insurance.


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Godwin, Ma'afu return for Force
Nick TaylorThe West Australian
Fri, 26 April 2013 9:08AM
Email Nick Taylor
Young centre Kyle Godwin and prop Salesi Ma'afu are set to return for Western Force when they take on Super Rugby ladder leaders ACT Brumbies in Canberra tomorrow.
Godwin, who was rested in the 22-16 loss to the Hurricanes after playing the previous eight games, is likely to come up against Christian Lealiifano.
Lealiifano, in his sixth season, will be out to prove to Wallabies coach Robbie Deans that, like Godwin, he can be a ball player from the centres ahead of the British Lions tour.
Godwin, 21, has attracted the interest of Deans, who said he had made "good strides" this season.
Ma'afu joined his teammates in Sydney before they travelled to Canberra yesterday.
He missed the Hurricanes clash with knee inflammation.
Scrum-half Ian Prior, who will move from the Brumbies to the Force next year, has been named on the bench to face his new club.
Brumbies coach Jake White said he expected a tough battle.
"As always with local derbies, we expect Saturday's match to be full of feeling and a tough contest right to the final siren," White said.
"The Force has a strong record against us here in Canberra. They almost got us here in round one last year and in 2011 they had a big come-from-behind win.
"They are a good team who have had some big wins over the last couple of weeks. It takes plenty of guts to overcome the Crusaders and the Reds, and we will need to be at the top of our game to get a win."
White has made one change to the starting line-up from the side that drew 19-all with Queensland Reds, Scott Sio coming in at loosehead prop.
Ben Alexander moves across the scrum to tighthead.
He has limited his player rotation. Wallabies inside centre Pat McCabe has not been recalled to the starting line-up, with fly-half Matt Toomua recovering from the concussion he suffered against the Reds.
White said he was happy with the combination of Toomua and Lealiifano.
"It's making sure that you're picking the team best equipped for the game. The guys are playing well in those positions," he said.
"As much as we have Patty pushing the door down as a Wallaby and wanting to play in the team, I think he'd appreciate, like all the players here do, that whoever is in has got to get the backing of the team.
"Scotty (Sio) has played really well when he has got some starts and it's a good week for him to get a crack.
"He has come along nicely this year and looks at home in the starting line-up."
Force coach Michael Foley will name his side today.
Brumbies *: Scott Sio, Stephen Moore, Ben Alexander, Peter Kimlin, Sam Carter, Ben Mowen, George Smith, Fotu Auelua, Nic White, Matt Toomua, Joseph Tomane, Christian Lealiifano, Tevita Kuridrani, Henry Speight, Jesse Mogg. *Reserves *: Siliva Siliva, Ruaidhri Murphy, Leon Power, Colby Faingaa, Ian Prior, Pat McCabe, Robbie Coleman.


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Solage can host weddings and receptions in its indoor or outdoor venues for up to 230 guests.
Carneros Resort and Spa features indoor and outdoor venues that can accommodate wedding ceremonies and receptions with up to 175 guests. Couples can choose from a ballroom, a hall, The Hilltop Dining Room, courtyards or The Arbor lawn. Additionally, the resort offers an elopment package, which includes officiating and photography services, bridal flowers, ceremony music and a sparkling wine toast.
There are six indoor and outdoor settings in which to host small intimate weddings. Indoors, the largest room, the Fountain View, can accommodate 110 guests. The Manor House Lawn is the largest outdoor venue and can hold 125 guests.
The Meritage has indoor and outdoor spaces, both intimate and grand, that make for a memorable wedding. Outdoor spaces include the Vineyard Deck and the Piazza Lawn, both of which have vineyard views. The resort has also its own wedding chapel, Our Lady of Grapes, which can hold 60 guests.
The Napa River Inn offers indoor and outdoor venues for intimate weddings. Couples can choose from historic indoor locations throughout the hotel or an outdoor courtyard setting. There are food and beverage services available through nearby local businesses, including Wine Valley Catering and Celadon Restaurant, plus custom cakes can be made by Sweetie Pies Bakery.
Couples can celebrate their special day at the Verasa in style. The Westin's variety of indoor and outdoor space can host wedding ceremonies, receptions and related events.
Couples can say their vows in an outdoor ceremony or indoor affair at the hotel, with views of the Napa River. The property offers various event spaces, including the River View Garden, River Terrace Patio and Terrace Court.
Guests can buy out all 20 rooms at the North Block for a completely personal affair. The hotel also caters to small elopement ceremonies and courtyard weddings. Rehearsal dinners, bridal showers and other celebrations can also be hosted at the hotel. Both indoor and outdoor spaces are available.


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Our Insights / Application Management
Application Management,Blog,Product Development,Resources 28th March 2019
Innovation Award for Intechnica's Work With Nisa
Julia Embury Senior Product Owner
The 'Direct to Store' (DTS) that Intechnica worked on in partnership with Nisa has won the Retail Systems Award for 'Online Innovation of The Year'.
The 'Direct to Store' was successfully launched at Nisa's retail exhibition in April. It allows independent retailers to trade with hundreds of locally sourced UK suppliers, search and compare product catalogues and order stock from multiple suppliers at the same time via one centralised online portal.
Peter Walker, Head of Business Systems at Nisa, said: "We are absolutely delighted to win the award and to receive recognition for the work that went into developing this system. Ultimately it streamlines processes for our retail partners and provides a better service to them, vastly reducing administration at store level and saving time."
The Retail Systems Awards 2018 awarded this years' accolade to Nisa's 'Direct to Store' on 27 June, just a couple of months after it was released, in the historic Waldorf Hilton in London.
Nisa and Intechnica beat fierce competition from John Lewis Partnership and ThoughtWorks, New Look and Criteo, Travis Perkins and Increasingly AI and more to lift the award for Online Innovation of the Year.
The 'Direct to Store' system simplified operations including invoicing, order processing and administration for Nisa members and as such has seen an increased uptake since the launch.
The Direct to Store Business Manager, Adrian Page, explained: "In addition to the range Nisa distributes centrally, independent retailers typically trade with hundreds of smaller and locally sourced suppliers. As the retailer must manually order and maintain product and price information directly with these suppliers, it requires a great deal of administration.
"Direct to Store removes this complication by providing a portal where all suppliers can list their product ranges, and an order portal for retailers to place orders quickly and efficiently. All product data is also available for retailers to download – simplifying the maintenance of product and price information."
Read more about Intechnica's work with Nisa in the case study: Nisa's Custom-Built B2B Platform Flawlessly Processes £1.4bn Orders.


Rock-Cable - For digital transmission of audio signals.
For digital transmission of audio signals.
Black Toslink cable, assembled with Toslink connectors with gold-plated contacts on both sides, and with PVC jacket and black chromed metal housings.
The Toslink cable is suitable for digital transmission of audio signals in consumer goods, e.g. for the transmission between CD/DVD players and audio amplifiers.


Horticulture has fascinated Charles Smith, Simply Green's founder, as long as he can remember. Simply Green was started in 2007 by John Willis and Charles Smith to bring a higher level of service to the local lawn fertilizer industry.
With a dedication to the customer, not a number on a sheet of paper at the end of the month. We built a foundation out of our company. We believe that as a service company, you only have to worry about that number when the quality of your service doesn't match the price you're charging.
It cost us more in fuel and time to run our program as we do, but we balance that by having no salesmen, and we don't advertise in the yellow pages. It doesn't pay in the end. The results our program produce, and the fact that our customers are our sales people because their neighbors are the customer we want. We believe that's the way to run a service company and keep it affordable and profitable.


text:
\section{Introduction and main results }
\labell{sec:main-results}
In this paper, we study the question of existence of leafwise
intersections for coisotropic submanifolds -- one of the
generalizations of the well-known Lagrangian intersection property to
coisotropic submanifolds. More specifically, we establish the
existence of leafwise intersections for restricted contact type
coisotropic submanifolds and Hamiltonian diffeomorphisms with Hofer
energy below a certain natural, sharp threshold. In other words, under
these conditions, we prove the existence of a leaf $F$ of the
characteristic foliation on a coisotropic submanifold $M$ such that
$\varphi(F) \cap F \neq \emptyset$ for a Hamiltonian diffeomorphism
$\varphi$.

The problem of existence of coisotropic intersections was first
addressed by Moser in the late 70s, \cite{Mo78}, and since then
various forms of the coisotropic intersection property have been
established.  For instance, the existence of ordinary intersections of
$M$ and $\varphi(M)$, or, roughly equivalently, lower bounds on the
displacement energy of $M$ have also been investigated. In this vein,
positivity of the displacement energy for stable coisotropic
submanifolds of ${\mathbb R}^{2n}$ was proved by Bolle, \cite{Bo96,Bo98}. (See
Section \ref{subsec:prelim-lw} for definitions.) In \cite{Gi07coiso},
revitalizing the subject a decade after Bolle's work, Ginzburg
extended Bolle's result to general symplectically aspherical
manifolds. This was generalized to closed, rational symplectic
manifolds by Kerman, \cite{Ke07coiso}, and, more recently, Usher,
\cite{U09coiso}, proved that the displacement energy of a stable
coisotropic submanifold of any closed or convex symplectic manifold is
positive.  Currently, the question is also being studied by Tonnelier,
\cite{To}.

The main result of this paper, falling in the realm of leafwise
intersections, is

\begin{Theorem}
\label{thm:lw}
Let $(W^{2n},\,\omega)$ be an exact symplectic manifold with
$c_1|_{\pi_2(W)} = 0$ which is geometrically bounded and wide. Let $M
\subset W$ be a closed, coisotropic submanifold of restricted contact
type. Then, for any compactly supported Hamiltonian diffeomorphism
$\varphi_H \colon W \to W$ with $\|H\| < \mathfrak{a}(M)$, there exists a leaf
$F$ of the characteristic foliation ${\mathcal F}$ on $M$ such that
$\varphi_H(F) \cap F \neq \emptyset$. Here,
$$
\mathfrak{a}(M) = \inf \{
A(\gamma) >0 \mid \gamma \text{ is a loop that is tangent to } {\mathcal F}
\text{ and contractible in } W \},
$$
where $A(\gamma)$ is the (negative) symplectic area bounded by
$\gamma$.
\end{Theorem}

\begin{Remark}
  Note that $\mathfrak{a}(M) >0$ whenever $M$ has restricted contact type; see
  \cite{Gi07coiso}. Furthermore, we set $\mathfrak{a}(M) = \infty$ if there is
  no $\gamma$ as in the definition of $\mathfrak{a}(M)$.
\end{Remark}

\begin{Remark}
  The assumption that $c_1|_{\pi_2(W)} = 0$ is not really essential in
  Theorem \ref{thm:lw} and included only for the sake of
  completeness. To be more precise, several of the results used in the
  proof of the theorem were originally established for symplectically
  aspherical manifolds. However, it is clear these results also hold
  for exact, wide, geometrically bounded manifolds without any
  additional assumptions on the first Chern class.
\end{Remark}

A symplectic manifold $W$ is said to be wide if it is open and admits
an arbitrarily large compactly supported Hamiltonian without
non-trivial contractible fast periodic orbits.  This notion was
introduced and discussed in detail in \cite{Gu07}. In particular,
examples of wide manifolds include manifolds that are convex at
infinity (e.g., ${\mathbb R}^{2n}$, cotangent bundles, Stein manifolds) and
twisted cotangent bundles. The essence of the wideness property lies
in the fact that on a wide manifold the top degree Floer homology is
non-zero for any non-negative compactly supported Hamiltonian which is
not identically zero. This allows one to construct an action selector
for geometrically bounded and wide manifolds, a tool utilized in the
proof of Theorem \ref{thm:lw}. (It is not known how to define action
selectors for an arbitrary geometrically bounded manifold.)

As has been pointed out above, the problem of leafwise intersections
was first considered in \cite{Mo78} where Moser established the
existence of leafwise intersections whenever $\varphi$ is $C^1$-close
to the identity and $M$ is simply connected.  The latter assumption
was removed by Banyaga, \cite{Ba}. Furthermore, Ekelund and Hofer,
\cite{EH,Ho90a}, showed that for contact type hypersurfaces in
${\mathbb R}^{2n}$, the $C^1$-smallness assumption on $\varphi$ can be replaced
by a much less restrictive hypothesis that the Hofer norm of $\varphi$
is smaller than a certain symplectic capacity of the domain bounded by
$M$. More recently, Ginzburg, \cite{Gi07coiso}, proved the existence
of leafwise intersections for restricted contact type hypersurfaces of
sub-critical Stein manifolds. The question has also been studied by
Ziltener, \cite{Zilt08}, under the additional assumption that the
characteristic foliation is a fibration.

Perhaps the most relevant to this work are recent papers
\cite{AF08,AF08generic} by Albers and Frauenfelder. In \cite{AF08},
the authors establish the existence of leafwise intersections for
hypersurfaces $M$ of restricted contact type, provided that the
ambient symplectic manifold is exact and convex at infinity and that
the Hofer norm of $\varphi$ does not exceed the minimal Reeb period
$\mathfrak{a}(M)$. In a follow-up work, \cite{AF08generic}, they prove that,
generically, the number of leafwise intersection points is at least
the sum of ${\mathbb Z}_2$-Betti numbers of $M$. The method used in
\cite{AF08,AF08generic} is the Rabinowitz Floer homology (see, e.g.,
\cite {CFO}). The result of the present paper is a generalization of
the theorem in \cite{AF08} to the case where $\mathit{codim}\, M > 1$, without
the multiplicity lower bound on the number of leafwise
intersections. Thus, the leafwise intersection property holds not just
for hypersurfaces, but is a feature of coisotropic submanifolds in
general.

\begin{Remark}
  Comparing Theorem \ref{thm:lw} with the result from \cite{Mo78,Ba},
  note that the latter holds for any coisotropic submanifold $M$, but
  the requirement on $\varphi$ is very restrictive. On the other hand,
  in Theorem \ref{thm:lw} the roles are reversed: we impose a strong
  condition on $M$ while the requirement on $\varphi$ has been
  relaxed. It is then natural ask whether Theorem \ref{thm:lw} can be
  generalized to any coisotropic submanifold $M$ along the lines of
  \cite{Mo78,Ba}. This, however, is not the case and leafwise
  intersections appear to be fragile:\emph{ There exists a
    hypersurface $M \subset {\mathbb R}^{2n}$ (diffeomorphic to
    $S^{2n-1}$) and a sequence of (autonomous) Hamiltonians $H_i
    \colon {\mathbb R}^{2n} \to {\mathbb R}$, supported within the same compact set,
    such that
\begin{itemize}
\item $\parallel H_i \parallel _{C^0} \to 0$, and
\item $\varphi_{H_i}(M)$ and $M$ have no leafwise intersections.
\end{itemize}}
\noindent This result will be proved in the forthcoming paper
\cite{Gu09cex}.
\end{Remark}

\subsection*{Acknowledgments}
The author is deeply grateful to Viktor Ginzburg for many useful
discussions and his numerous valuable remarks and suggestions.

\section{Preliminaries}
\label{sec:lw-int}
We start this section by recalling the relevant definitions and basic
results concerning coisotropic submanifolds.  In Section
\ref{subsec:conv}, we set our conventions and notation.  We recall the
definition of the action selector for wide manifolds and state its
relevant properties in Section \ref{subsec:selector}.

\subsection{Preliminaries on coisotropic submanifolds}
\label{subsec:prelim-lw}
Let $(W^{2n}, \omega)$ be a symplectic manifold and let $M\subset W$
be a closed, coisotropic submanifold of codimension $k$. Set
$\wM=\omega|_M$. Then, as is well known, the distribution $\ker \wM$
has dimension $k$ and is integrable. Denote by ${\mathcal F}$ the
characteristic foliation on $M$, i.e., the $k$-dimensional foliation
whose leaves are tangent to the distribution $\ker \wM$.

\begin{Definition}
\label{def:B}
The coisotropic submanifold $M$ is said to be \emph{stable} if there
exist one-forms $\alpha_1,\ldots,\alpha_k$ on $M$ such that $\ker
d\alpha_i\supset \ker\wM$ for all $i=1,\ldots,k$ and
\begin{equation*}
\labell{eq:ct}
\alpha_1\wedge\cdots\wedge\alpha_k\wedge \wM^{n-k}\neq 0
\end{equation*}
anywhere on $M$. We say that $M$ has \emph{contact type} if the forms
$\alpha_i$ can be taken to be primitives of $\wM$. Furthermore, $M$
has \emph{restricted} contact type if the forms $\alpha_i$ extend to
global primitives $\bar{\alpha}_i$ of $\omega$ on $W$.
\end{Definition}

Stable and contact type coisotropic submanifolds were introduced by
Bolle in \cite{Bo96,Bo98} and considered in a more general setting by
Ginzburg, \cite{Gi07coiso}, and afterwards also by Kerman,
\cite{Ke07coiso}, and by Usher, \cite{U09coiso}. (See also \cite{Dr}.)
Referring the reader to \cite{Gi07coiso} for a discussion of the
requirements of Definition \ref{def:B} and several illustrating
examples, let us merely note that these requirements are natural but
quite restrictive. For example, a stable Lagrangian submanifold is
necessarily a torus. In this paper, we will be mainly concerned with
coisotropic submanifolds of restricted contact type. This condition is
a generalization of its namesake for hypersurfaces, as is the case for
contact type or stability conditions.

Assume that $M$ is stable. The normal bundle to $M$ in $W$ is trivial
since it is isomorphic to $T^*{\mathcal F}$ and, thus it can be identified with
$M \times {\mathbb R}^k$. From now on we identify a small neighborhood of $M$
in $W$ with a neighborhood of $M$ in $T^*{\mathcal F} = M \times {\mathbb R}^k$ and use
the same symbols $\wM$ and $\alpha_i$ for differential forms on $M$
and for their pullbacks to $M \times {\mathbb R}^k$. (Thus, we will be
suppressing the pullback notation $\pi^*$, where $\pi \colon M \times
{\mathbb R}^k \to M$, unless its presence is absolutely necessary.)  Using the
Weinstein symplectic neighborhood theorem, we then have

\begin{Proposition}[\cite{Bo96, Bo98}]
\label{prop:normalform}
Let $M$ be a closed, stable coisotropic submanifold of $(W^{2n},
\omega)$ with $ \mathit{codim}\, M = k$.  Then, for a sufficiently small $r>0$,
there exists a neighborhood of $M$ in $W$ which is symplectomorphic to
$U_r=\{ (q,p) \in M \times {\mathbb R}^k \mid |p| < r \}$ equipped with the
symplectic form $\omega = \wM + \sum_{j=1}^k d(p_j \alpha_j)$. Here
$(p_1,\ldots,p_k)$ are the coordinates on ${\mathbb R}^k$ and $|p|$ is the
Euclidean norm of $p$.
\end{Proposition}

As an immediate consequence of Proposition \ref{prop:normalform}, we
obtain a family of coisotropic submanifolds $M_p = M\times \{p\} $,
for $p \in B^k_r$, of $W$ which foliate a neighborhood of $M$ in $W$,
where $B_r^k$ is the ball of radius $r$, centered at the origin in
${\mathbb R}^k$. Moreover, a leaf of the characteristic foliation on $M_p$
projects onto a leaf of the characteristic foliation on $M$.

Furthermore, we have

\begin{Proposition}[\cite{Bo96, Bo98, Gi07coiso}]
\label{prop:flow}
Let $M$ be a stable coisotropic submanifold.

\begin{enumerate}

\item[(i)] The leafwise metric $(\alpha_1)^2+\cdots+(\alpha_k)^2$ on
  ${\mathcal F}$ is leafwise flat.

\item[(ii)] The Hamiltonian flow of
  $\rho=(p_1^2+\cdots+p_k^2)/2=|p|^2/2$ is the leafwise geodesic flow
  of this metric.

\end{enumerate}

\end{Proposition}

\begin{Remark}
  Using this property, it is not hard to show that, whenever $M$ has
  restricted contact type, for every leaf $F$ of ${\mathcal F}$, the kernel of
  the map $ H_1(F;{\mathbb R}) \to H_1(W;{\mathbb R})$ is either trivial or one
  dimensional. For instance, if $H_1(W;{\mathbb R})=0$, every leaf $F$ is
  diffeomorphic to either ${\mathbb R}^k$ or $S^1 \times {\mathbb R}^{k-1}$.
\end{Remark}

\subsection{Conventions and notation}
\label{subsec:conv}
In this section we set our conventions and notation.

Let $(W^{2n},\omega)$ be a symplectically aspherical manifold, i.e.,
$\omega|_{\pi_2(W)}=c_1|_{\pi_2(W)}=0$.  Denote by $\Lambda W$ the
space of smooth contractible loops $\gamma\colon S^1\to W$ and
consider a time-dependent Hamiltonian $H\colon S^1\times W\to {\mathbb R}$,
where $S^1={\mathbb R}/{\mathbb Z}$.  Setting $H_t = H(t,\cdot)$ for $t\in S^1$, we
define the action functional $A_H\colon \Lambda W\to {\mathbb R}$ by
$$
A_H(\gamma)=A(\gamma)+\int_{S^1} H_t(\gamma(t))\,dt,
$$
where $A(\gamma)$ is the negative symplectic area bounded by $\gamma$, i.e.,
$$
A(\gamma)=-\int_{\bar{\gamma}}\omega,
$$
where $\bar{\gamma}\colon D^2\to W$ is such that
$\bar{\gamma}|_{S^1}=\gamma$.

The least action principle asserts that the critical points of $A_H$
are exactly contractible one-periodic orbits of the time-dependent
Hamiltonian flow $\varphi_H^t$ of $H$, where the Hamiltonian vector
field $X_H$ of $H$ is defined by $i_{X_H}\omega=-dH$. We denote the
collection of such orbits by ${\mathcal P}_H$. The action spectrum ${\mathcal S}(H)$ of
$H$ is the set of critical values of $A_H$. In other words, ${\mathcal S}(H)=\{
A_H(\gamma)\mid \gamma\in {\mathcal P}_H\}$. This is a zero measure set; see,
e.g., \cite{HZ94}.

In what follows we will always assume that $H$ is compactly supported
and set ${\mathit supp}\, H=\bigcup_{t\in S^1}{\mathit supp}\, H_t$.  In this case, ${\mathcal S}(H)$
is compact and hence nowhere dense.

Let $J=J_t$ be a time-dependent almost complex structure on $W$. A Floer
anti-gradient trajectory $u$ is a map $u\colon {\mathbb R}\times S^1\to W$
satisfying the equation
\begin{equation*}
\label{eq:floer}
\frac{\partial u}{\partial s}+ J_t(u) \frac{\partial u}{\partial t}=-\nabla H_t(u).
\end{equation*}
Here, the gradient is taken with respect to the time-dependent
Riemannian metric $\omega(\cdot,J_t\cdot)$. Denote by $u(s)$ the curve
$u(s,\cdot)\in \Lambda W$.

The energy of $u$ is defined as
\begin{equation*}
\label{eq:energy}
E(u)
=
\int_{-\infty}^\infty \left\|\frac{\partial u}{\partial s}\right\|_{L^2(S^1)}^2\,ds
=
\int_{-\infty}^\infty \int_{S^1}\left\|\frac{\partial u}{\partial t}-J\nabla H (u)
\right\|^2 \,dt\,ds.
\end{equation*}
We say that $u$ is asymptotic to $x^\pm\in {\mathcal P}_H$ as $s\to\pm \infty$
or connecting $x^-$ and $x^+$ if $\lim_{s\to\pm\infty} u(s)=x^\pm$ in
$\Lambda W$. More generally, $u$ is said to be partially asymptotic to
$x^\pm\in {\mathcal P}_H$ at $\pm\infty$ if $u(s^{\pm}_n)\to x^\pm$ for some
sequences $s^{\pm}_n\to\pm\infty$. In this case
$$
A_H(x^-)-A_H(x^+)=E(u).
$$

In this paper the manifold $W$ is assumed to be exact and, hence,
open. In this case, in order for the Floer homology to be defined, we
assume that $W$ is geometrically bounded. This assumption gives us
sufficient control of the geometry of $W$ at infinity which is
necessary in the case of open manifolds. Examples of such manifolds
include symplectic manifolds that are convex at infinity (e.g.,
${\mathbb R}^{2n}$, cotangent bundles) as well as twisted cotangent
bundles. (See, e.g., \cite{AL,CGK} for the precise definition and a
discussion of geometrically bounded manifolds.) Under the hypotheses
that $W$ is symplectically aspherical and geometrically bounded, the
compactness theorem for Floer's connecting trajectories holds and the
filtered ${\mathbb Z}$-graded Floer homology of a compactly supported
Hamiltonian on $W$ is defined; see, e.g., \cite{CGK,GG04}.

\subsection{Action selector for wide manifolds}
\label{subsec:selector}
In this section we briefly recall the definition and relevant
properties of the action selector defined in \cite{Gu07} for wide and
geometrically bounded manifolds. We refer the reader to \cite{Gu07,
  Gi07coiso} for more details. Here we only note that action selectors
were constructed in \cite{schw00} for closed manifolds and in
\cite{HZ94} and \cite{Vi92} for ${\mathbb R}^{2n}$ and cotangent bundles,
respectively. The approach of \cite{schw00} has been extended to
manifolds convex at infinity in \cite{FS03}.

\emph{The definition.}  Let $H\colon S^1 \times W\to {\mathbb R}$ be a
compactly supported, non-negative Hamiltonian which is not identically
zero. As was proved in \cite{Gu07}, on a wide manifold, the top degree
Floer homology of $H$ for the action interval $(0,\infty)$ is non-zero
and it carries a canonically defined homology class. (Note that the
homology group itself depends on $H$.) Call this class $[\max H]$ and
define
$$
\sigma(\varphi_H)=\inf\{a> 0\mid j^a_H\left([\max H]\right)=0\}\in
  {\mathcal S}(H),
$$
where
$$
j^a_H\colon \HF^{(0,\,\infty)}(H)\to \HF^{(a,\,\infty)}(H)
$$
is the quotient map. (This definition coincides with the one from
\cite{FS03} whenever $W$ is convex.)

\emph{Properties of the action selector.}  Focusing on the ones that
are relevant for what follows, recall that the action selector
$\sigma$, defined as above, has the following properties for
non-negative Hamiltonians:

\begin{enumerate}

\item[(S1)] $\sigma$ is monotone, i.e.,
  $\sigma(\varphi_K)\leq\sigma(\varphi_H)$, whenever $0\leq K\leq H$
  point-wise;

\item[(S2)] $0\leq\sigma(\varphi_H)\leq E^+(H)$ for any $H\geq 0$,
  where
$$
E^+(H)=\int_{S^1}\max_W H_t\,dt;
$$

\item[(S3)] $\sigma(\varphi_H)>0$, provided that $H\geq 0$ is not
  identically zero;

\item[(S4)] $\sigma(\varphi_H)$ is continuous in $H$ in the
  $C^0$-topology.

\end{enumerate}
We refer the reader to \cite{Gu07} for the proofs of these properties.

\section {Proof of Theorem \ref{thm:lw}}
\label{sec:lw-proof}
In this section we prove Theorem \ref{thm:lw}. Throughout the proof,
as in Section \ref{subsec:prelim-lw}, a neighborhood of $M$ in $W$ is
identified with a neighborhood of $M$ in $M\times {\mathbb R}^k$ equipped with
the symplectic form $\omega = \wM + \sum_{j=1}^k d(p_j \alpha_j)$.
Using this identification, we denote by $U_R$, with $R>0$ sufficiently
small, the neighborhood of $M$ in $W$ corresponding to $M\times
B^k_R$. (Thus, $U_R=\{\rho < R^2 /2\}$.)

\begin{proof}[Proof of Theorem \ref{thm:lw}]
  First note that, without loss of generality, we may assume that
  $\operatorname{Fix}(\varphi_H) \cap M = \emptyset$, for otherwise the assertion is
  obvious. Then $\varphi_H$ has no fixed points near $M$, say in a
  tubular neighborhood $U=M \times B_R^k$ of $M$ in $W$, where $R>0$
  is sufficiently small. Let $U_r \subset U$ be a smaller tubular
  neighborhood of $M$ for some $r<R$. We may also require that $H \geq
  0$ and $\| H\|= E^+(H)$. (This can be achieved by replacing $H$ by
  $f \cdot (H-\min H)$, where $f$ is a cut-off function equal to one
  near ${\mathit supp}\,(H) \cup \bar{U}$.)

  Let $K\geq 0$ be a non-negative function on $[0,r]$ such that

\begin{itemize}
\item $K(0) = \max K >0$ and $K \equiv 0 $ near $r$;
\item $K$ is strictly decreasing until it becomes zero;
\item all odd-order derivatives of $K$ vanish at $0$, and $K''(0)<0$
is close to zero.
\end{itemize}
Abusing notation, we also denote by $K$ the function on $W$, equal to
$K(|p|)$ on $U_r$, where $ |p| = \sqrt{2 \rho}$, and extended to be
identically zero outside $U_r$. Note that $K$ on $W$ has only two
critical values: $\max K$ and $0$. Observe also that the Hamiltonian
flow of $K$ on $U_r$ is just a reparametrization of the leafwise
geodesic flow on $M$ and the flow is the identity map outside $U_r$.

Recall that $\mathfrak{a}(M) = \inf \{ A(\gamma) >0 \mid \gamma \text{ is
  tangent to } {\mathcal F} \text{ and contractible in } W \} > 0$, where
$A(\gamma)$ denotes the negative symplectic area bounded by the orbit
$\gamma$, and set $\mathfrak{a}(M) = \infty$ if there is no such $\gamma$.

The first ingredient in the proof of Theorem \ref{thm:lw} is

\begin{Lemma}
\label{lemma:K}
In the above setting, there exists a constant $C>0$, depending on
$U_r$ but not on $K$, such that whenever $\max K \geq C$, we have
$\sigma(\varphi_K) \geq C$. Moreover, $C$ can be taken to be of the
form $C=(1-O(r)) \mathfrak{a}(M)$. If $\mathfrak{a}(M)=\infty$, the constant $C$ can be
taken to be arbitrarily large.
\end{Lemma}

\begin{Remark}
  A version of this lemma also holds when $M$ is just stable although
  in this case we cannot guarantee that $C$ has the desired form.
  However, one can then take $C$ to be independent of $r$ (c.f.\
  Theorem 2.7 of \cite{Gi07coiso}).
\end{Remark}

\begin{proof}[Proof of Lemma \ref{lemma:K}]
  Set $ T_x^\perp {\mathcal F} = \cap \ker {\alpha_j} \subset T_{(x,p)}
  M_p$. Then,
\begin{equation}
\labell{eq:decom}
T_{(x,p)}U = T_x^\perp {\mathcal F}  \times \left( T_x {\mathcal F} \times T_p B \right)
\end{equation}
is a decomposition of the tangent space at a point $(x,p) \in U$ into
a direct sum of two symplectic subspaces.  Indeed, observe that
$$
\omega_{(x,p)}= \left(1 + \sum_{j=1}^k p_j \right) {\omega_x}|_{T_x^\perp {\mathcal F}}
+ \sum_{j=1}^k dp_j \wedge \alpha_j,
$$
where the first symplectic form vanishes on $T_x {\mathcal F} \times T_p B$ and
the second one vanishes on $ T_x^\perp {\mathcal F} $.

Let us now introduce an almost complex structure $J = J(x,p)$ on $W$,
compatible with $\omega$ and \eqref{eq:decom}. To this end, observe
that the non-degeneracy condition on $\alpha_i$'s implies that the
forms are linearly independent and leafwise closed. Let
$X_1,\ldots,X_k$ be the basis in $T_x {\mathcal F}$ dual to
$\alpha_1,\ldots,\alpha_k$, i.e., $\alpha_i(X_j)=\delta_{ij}$.
Clearly, $\partial_{p_1},\ldots,\partial_{p_k}$ form a basis in $T_p
B$.  Define $J$ such that $ T_x^\perp {\mathcal F} $ and $T_x {\mathcal F} \times T_p B$
are complex subspaces in $T_{(x,p)}U$; furthermore, $J|_{T_x^\perp
  {\mathcal F}}$ is compatible with ${\omega_x}|_{T_x^\perp {\mathcal F}}$; and on $T_x
{\mathcal F} \times T_p B$, we have $J(X_i)=\partial_{p_i}$. (Outside $U_R$, we
take $J$ to be an arbitrary almost complex structure compatible with
$\omega$.)  The pair $(\omega,\,J)$ gives rise to a metric on $M$
compatible with $\omega$ and such that \eqref{eq:decom} is an
orthogonal decomposition.

With respect to this metric, $\parallel
\partial_{p_i} \parallel = 1$ and $\parallel X_i \parallel =
1$. Moreover, $\parallel \omega \parallel := \sup_{X,Y}\,\omega (X,Y)
=1$, where the supremum is taken over all tangent vectors $X,\,Y$ with
norm one. Note also that the Hamiltonian vector field of
$\rho=(p_1^2+\cdots+p_k^2)/2$ is $X_\rho=\sum_{j=1}^k p_j X_j$ and,
thus, $J(X_\rho)= \sum_{j=1}^k p_j \partial_{p_j} $.

We may assume without loss of generality that the forms $\alpha_i$
extend to primitives $\bar{\alpha}_i$ of $\omega$ on $W$ such that
$$
\bar{\alpha}_i = \alpha_i + \sum_{j=1}^k p_j {\alpha}_j  \mbox{ on } U_R.
$$
In the spirit of \cite{Bo96, Bo98, Gi07coiso}, our next goal is to
define smooth one-forms $\beta_i$ on $W$ for $i=1,\ldots,k$ such that
$ \beta_i$ agrees with $\alpha_i$ on $U_r$ and with $\bar{\alpha}_i$
outside $U_R$ and that
$$
\parallel d\beta_i \parallel_{C^0} \leq 1 + O(r).
$$
To this end, let $g=g_r(y)$ be a family of smooth, non-negative,
monotone increasing functions defined on $[0,\infty)$ such that
$g\equiv 0$ on $[0,r]$ and $g\equiv 1$ on $[R,\infty)$, and that $0
\leq g'(y) \, y + g(y) \leq 1+ O(r)$. It is not hard to write an
explicit formula for such functions $g$. Abusing notation, denote also
by $g$ the function $g(|p|)$ defined on $W$. Finally, define the
one-forms $\beta_i$ as
$$
\beta_i =
\begin{cases}
\alpha_i + g(|p|) \sum_{j=1}^k p_j {\alpha}_j & \mbox{on } U_R\\
\bar{\alpha}_i & \mbox{outside } U_R
\end{cases}
$$
Now a straightforward, but tedious, calculation shows that $
\parallel d\beta_i \parallel_{C^0} \leq 1 + O(r)$. It is particularly
easy to see that this is the case when $M$ is a hypersurface. Namely,
then $k=1$ and $d\beta = (1+f) d\alpha + df \wedge \alpha$, where
$f(y)=y\,g(y)$. Notice that we only need to prove the desired estimate
on $U_R$. Then, $d\alpha = \pi^* \wM$ and the first form vanishes on
$T_x {\mathcal F} \times T_p B$, while the second form vanishes on $T_x^\perp
{\mathcal F} = \ker \alpha$. Thus, it suffices to prove the estimate on these
two symplectic subspaces separately. To this end, observe that on
$U_R$ we have
$$
\parallel d{\beta}|_{T_x^\perp {\mathcal F}} \parallel = (1+f)/(1+p) = 1 + O(r),
$$
as is easy to see. On the other hand, $T_x {\mathcal F} \times T_p B$ is
spanned by $\{ X, \partial_p \}$ and
$$
\parallel d{\beta}|_{T_x {\mathcal F} \times T_p B} \parallel = |d \beta (X, \partial_p)|.
$$
By the definition of $g$, and, hence, of $f$, we have $d \beta (X,
\partial_p) = f' \leq 1 +O(r)$. The calculation in the general case is more
involved but follows the same track as the one for hypersurfaces.

A feature of the form $\beta_i$, important in what follows, is that
\begin{equation}
\label{eq:0}
i_{X_K}d\beta_i=0.
\end{equation}
To see this, observe first that \eqref{eq:0} trivially holds outside
$U_r$, for ${\mathit supp}\, K \subset U_r$.  Moreover, $d\beta_i=d \pi^*
\alpha_i = \pi^* \wM$ on $U_r$ and we then have
$$
i_{X_K}d\beta_i=i_{X_K}\pi^*d\alpha_i=K'(\rho)\,i_{\pi_*X_\rho}d\alpha_i=0.
$$
The last equality follows from the fact that $\pi_*X_\rho$ is tangent
to ${\mathcal F}$ by Proposition~\ref{prop:flow} and that $T{\mathcal F} \subset \ker
d\alpha_i$ since $M$ is stable.

Let $\epsilon >0$ be small enough so that $f=\epsilon K$ is
$C^2$-small and consider a linear homotopy from $K$ to $f$, running
through functions of $\rho$.  Then, there exist an orbit $\gamma$ of
$K$ and a homotopy trajectory $u$ partially asymptotic to $\gamma$ at
$-\infty$ and to a critical point of $f$ (and, hence, of $K$) on $M$
at $\infty$ such that
\begin{itemize}
\item[(i)] $A_K(\gamma) \leq \sigma(\varphi_K)$, and

\item[(ii)] $E(u) \leq A_K(\gamma) - \epsilon \max K \leq
  \sigma(\varphi_K) - \epsilon \max K$.
\end{itemize}
This fact is essentially a particular case of Proposition 5.4 in
\cite{Gi07coiso} with the displaceability requirement being
inessential for us and it can be proved exactly in the same way.

We claim that
\begin{equation}
\labell{eq:energy-bound}
E(u)
  \geq \parallel d\beta_i \parallel^{-1} \left|\int_{\pi(\gamma)}\alpha_i\right|.
\end{equation}
To prove \eqref{eq:energy-bound}, fix $s_n^\pm\to \pm\infty$ such that
$u(s_n^+)$ converges to
a point of $M$ and $u(s_n^-)$ converges to $\gamma$ in
$C^\infty(S^1,W)$. Then, using \eqref{eq:0}, we have

\begin{eqnarray*}
E(u)
&=&
\int_{-\infty}^\infty \int_{S^1}
\left\| \frac{\partial u}{\partial s}\right\|
\cdot
\left\| \frac{\partial u}{\partial t}-X_K\right\|
\,dt\,ds
\\
&\geq&
\parallel d\beta_i \parallel^{-1}
\int_{-\infty}^\infty \int_{S^1}
\left|d\beta_i\left(\frac{\partial u}{\partial s}, \frac{\partial u}{\partial t}-X_K\right)\right|
\,dt\,ds
\\
&\geq&
\parallel d\beta_i \parallel^{-1}
\liminf_{n\to\infty}
\left|\int_{s_n^-}^{s_n^+}\int_{S^1}
d\beta_i\left(\frac{\partial u}{\partial s}, \frac{\partial u}{\partial t}-X_K\right)
\,dt\,ds\right|
\\
&=&
\parallel d\beta_i \parallel^{-1}
\liminf_{n\to\infty}
\left|\int_{s_n^-}^{s_n^+}\int_{S^1}
d\beta_i\left(\frac{\partial u}{\partial s}, \frac{\partial u}{\partial t}\right)
\,dt\,ds\right|.
\end{eqnarray*}
By Stokes' formula,
$$
\left|\int_{s_n^-}^{s_n^+}\int_{S^1}
d\beta_i\left(\frac{\partial u}{\partial s}, \frac{\partial u}{\partial t}\right)
\,dt\,ds\right|
=
\left|\int_{u(s_n^+)}\beta_i-\int_{u(s_n^-)}\beta_i\right|
\longrightarrow
\left|\int_{\gamma}\beta_i\right|
$$
as $n\to\infty$.
Furthermore, recall that $\gamma$ is contained in $U_r$
and $\beta_i|_{U_r}=\pi^*\alpha_i$. Thus,
$$
\left|\int_{\gamma}\beta_i\right|=\left|\int_{\gamma}\pi^*\alpha_i\right|
=\left|\int_{\pi(\gamma)}\alpha_i\right|,
$$
which completes the proof of \eqref{eq:energy-bound}.

Set $C= \parallel d\beta_i \parallel^{-1} \mathfrak{a}(M) $, assuming that
$\mathfrak{a}(M)< \infty$. It is clear that then $C =\left( 1 - O(r) \right)
\mathfrak{a}(M) >0$. Assume now that $\max K \geq C$ and recall that $\sigma
(\varphi_K) \leq \max K$.  If $\sigma(\varphi_K) = \max K$, the proof
is finished. Let us focus on the case when $\sigma(\varphi_K) < \max
K$. Then, $\gamma \subset U_r$ is necessarily nontrivial.  Indeed,
trivial orbits of $K$ occur where $K$ is constant, i.e. $K=0$ or $K =
\max K$. Since (i) implies that $A_K(\gamma) \leq \sigma(\varphi_K) <
\max K$, we must have $K = 0$ whenever $\gamma$ is constant and,
hence, $A_K(\gamma)=0$. Consequently, we infer from (ii) that $E(u) <
0$, which is clearly a contradiction.  Hence, $\gamma$ is nontrivial
and, as an immediate consequence, we have
$\left|\int_{\pi(\gamma)}\alpha_i\right| > 0$.  Then, by definition of
$\mathfrak{a}(M)$,
\begin{equation*}
\sigma(\varphi_K)
\geq
E(u)
\geq
\parallel d\beta_i \parallel^{-1} \left|\int_{\pi(\gamma)}\alpha_i\right|
\geq
\parallel d\beta_i \parallel^{-1} \mathfrak{a}(M)
=
C.
\end{equation*}

Finally, note that if $\mathfrak{a}(M)= \infty$, we must necessarily have
$\sigma(\varphi_K) = \max K$ and, then, $C$ can be taken to be
arbitrarily large.
\end{proof}

Returning to the proof of Theorem \ref{thm:lw}, let $C >0$ be a
constant as in Lemma \ref{lemma:K} and assume from now on that the
function $K$ has the additional property that $\max K = 2 C >
C$. Thus, $\sigma(\varphi_K) \geq C$.

For the sake of convenience, let us reparametrize $H$ and $K$, as
functions of $s$ and $t$, so that $H_t \equiv 0$ on $[0,1/2]$ and $K_t
\equiv 0$ on $[1/2,1]$. (Note that the time-one maps, the Hofer norms,
the action spectra, and the action selectors remain the same.) Then,
the flow given by (a smooth reparametrization of) the concatenation of
paths $\varphi_K^t$ and $\varphi_H^t\varphi_K$, where $t \in [0,1]$,
is homotopic with fixed end points to $\varphi_H^t\varphi_K^t$, and is
generated by the Hamiltonian $H_t+K_t$.

Consider the family of diffeomorphisms
$\psi_s^t=\varphi_H^t\varphi_{sK}^t$ for $s\in [0,\,1]$, starting at
$\psi_0^t=\varphi_H^t$ and ending at
$\psi_1^t=\varphi_H^t\varphi_{K}^t$. Note that, for a fixed $s$, we
have $\psi_s ^0 = {\mathit id}$ and $\psi_s^1=\psi_s=\varphi_H \varphi_{sK}$,
and, thus each $\psi_s^t$, as an an element of $\Hamt$, is generated
by the Hamiltonian
$H_t + sK_t =: G_t^{s}$.
Examining the fixed points of $\psi_s$, as is easy to see using the
assumption that $\operatorname{Fix}(\varphi_H) \cap U_r = \emptyset$, we have the
disjoint union decomposition
$$
\operatorname{Fix}(\psi_s)=\operatorname{Fix}(\varphi_H)\sqcup Z_s, \text{ where } Z_s=\{x\in U_r
\mid \varphi_H^{-1}(x)=\varphi_{sK}(x)\}.
$$
Furthermore, $\sigma(\psi_s)$, as a function of $s$, is not
constant. Indeed, first note that, by our assumption on $H$,
$$\sigma(\psi_0) = \sigma(\varphi_H)\leq \|H\| < C. $$
Since $H\geq 0$ and $\sigma$ is monotone, we also have the inequality
$$
\sigma(\psi_1)=\sigma(\varphi_H\varphi_K) \geq \sigma(\varphi_K).
$$
Finally, using Lemma \ref{lemma:K} along with these two inequalities,
we obtain
$$
\sigma(\psi_0) < C \leq \sigma(\varphi_K) \leq \sigma(\psi_1).
$$
This shows that $\sigma(\psi_s)$ is not constant. On the other hand,
$\sigma(\psi_s)$ is a continuous function of $s$ and the set $\operatorname{Fix}
(\varphi_H)$ is nowhere dense. Thus, $Z_{s_0} \neq \emptyset$ for some
$s_0 \in (0,1]$ and $\sigma(\psi_{s_0})$ is the action value of
$\psi_{s_0}$ on a fixed point $x \in Z_{s_0}$. (Notice that $Z_0= \operatorname{Fix}
(\varphi_H) \cap U_r = \emptyset$.)

Let $\gamma(t)$ be the one-periodic orbit of the time-dependent flow
$\psi_{s_0}^t = \varphi_H^t \varphi^t_{s_0K}$, passing through
$x=\gamma(0)$. This flow is generated by $G^{s_0}_t= H_t + s_0 K_t$
and, due to the above parametrizations of $H$ and $K$, the orbit
$\gamma$ is comprised of two parts: $\gamma_K(t)=\varphi^t_{s_0K}(x)$
-- a trajectory of $s_0K$ beginning at $x$ and ending at
$y=\varphi_{s_0K}(x)$ -- and $\gamma_H(t)=\varphi_H^t(y)$ -- a
trajectory of $H$ beginning at $y$ and ending at $x$. Note that $x$
and $y$ lie on the same leaf of the characteristic foliation on some
$M_p$ and $\varphi_H(y)=x$, where $p \in B_r^k$ such that $0<|p|<r$.

Furthermore,
\begin{equation*}
\label{eq:gamma}
\sigma(\psi_{s_0})=A_{G^{s_0}}(\gamma) =-\int_{\gamma_K}\lambda+
\int_0^{1/2} s_0K_t(\gamma_K(t))\,dt + A_H(\gamma_H),
\end{equation*}
where $\lambda$ is a global primitive of $\omega$. Our next goal is to
show that
\begin{equation}
\label{eq:T}
\left|\int_{\gamma_K}\lambda \right|
=
\left| -\sigma(\psi_{s_0})+
\int_0^{1/2}s_0K_t(\gamma_K(t))\,dt + A_H(\gamma_H) \right|
\end{equation}
is bounded by a constant independent of $K$, which will in turn imply
that the time required to move $x$ to $y$ is (uniformly) bounded. To
see this, first recall that
$$
0 < \sigma(\psi_{s_0}) \leq \| H_t + s_0 K_t \| \leq \|H\| + \|K\|
\leq C + 2C = 3C.
$$
The second term in \eqref{eq:T} is bounded from below by zero since
$K\geq 0$ and from above by $\max K/2 = C$. Finally,
$|A_H(\gamma_H)|$ is bounded. Indeed, consider the function
$h(z)=A_H(\varphi_H^t(z))$, where $t\in [0,1]$, defined using the
primitive $\lambda$. (For example, $h(y)=A_H(\gamma_H)$.) This
function is compactly supported and independent of $K$. Letting $C' =
\max \{|\max h|,|\min h|\}$, a constant independent of $K$, we see
that $|\int_{\gamma_K}\lambda| \leq 4 C + C'$.

Let $X=X_{\rho}/|p|$ and note that $\alpha_i (X)=p_i/|p|$, where
$|p|=\sqrt{2\rho}$. Consider the primitives $\lambda = \bar{\alpha}_i$
of $\omega$ for $i=1,\ldots,k$.  As was shown above,
$$
\left| \int_{\gamma_K}\lambda \right|
=
\left| \int_{\gamma_K}\bar{\alpha}_i \right|
\leq
C_i,
$$
where $C_i >0$ is a constant independent of $K$. On the other hand,
on $U_R$ we have
$$
\bar{\alpha}_i (X) =
\alpha_i (X) + \sum_{j=1}^k p_j \alpha_j (X) = \frac{p_i}{|p|} + |p|.
$$
As a result,
$$
\int_{\gamma _K} \bar{\alpha}_i =
\int_0^T \bar{\alpha}_i (X) \, dt =
T \left( \frac{p_i}{|p|} + |p| \right),
$$
where $T$ is the time required for the flow of $X$ to move $x$ to $y$.
For $\bar{C} = \sum_{i=1}^k C_i$, we then have
\begin{eqnarray*}
\bar{C} \geq
\sum_{i=1}^k \left| \int_{\gamma _K} \bar{\alpha}_i \right|
&=&
\sum_{i=1}^k T \left| \frac{p_i}{|p|} + |p| \right|
\\
&\geq& T \left( \sum_{i=1}^k \frac{|p_i|}{|p|}  -
k \, |p| \right)
\\
&\geq& T \left( c - k \, |p| \right),
\end{eqnarray*}
where $ c= \min_{|p|=1} \sum_{i=1}^k |p_i| >0 $. Note that $|p| <
r$. Thus, when $r < c/ 2k$, we have $ \bar{C} \geq T \, c /2$.  Hence
$ T \leq 2 \bar{C}/c $, where the right-hand side is independent of
$K$.

As the final step of the proof, consider a sequence $r_i \to 0$ and a
sequence of Hamiltonians $K_i$ such as $K$, supported on the (smaller
and smaller) neighborhood $U_{r_i}$ of $M$. For each $K_i$, we have a
pair of points $x_i$ and $y_i$ lying on the same leaf of the
characteristic foliation on some $M_{p(i)}$ with $|p(i)|\to 0$ and
such that $\varphi_H(y_i)=x_i$. Furthermore, the time $T_i$ required
to move $x_i$ to $y_i$ is bounded from above by a constant independent
of $K_i$ and $r_i$. Applying the Arzela--Ascoli theorem and passing if
necessary to a subsequence, we obtain points $x=\lim x_i$ and $y=\lim
y_i$ on $M$ lying on the same leaf of the characteristic foliation on
$M$ and such that $\varphi_H(y)=x$.  This completes the proof of
Theorem \ref{thm:lw}.

\end{proof}

redpajama_set_name: RedPajamaArXiv


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