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There have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past 25 to 30 years. Many of the researchers who have produced these excellent results were inspired by this little book - or by Fred Almgren himself. The book is indeed a delightful invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behavior of soap films.When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This new tool makes possible much exciting new analysis and many new results. Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. The author's hope in writing this book was to encourage young mathematicians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well.
This volume consists of papers written by eminent scientists from the international mathematical community, who present the latest information concerning the problem of Plateau after its classical solution by Jesse Douglas and Tibor Rad¢. The contributing papers provide insight and perspective on various problems in modern topics of Calculus of Variations, Global Differential Geometry and Global Nonlinear Analysis as related to the problem of Plateau.
The most immediate one-dimensional variation problem is certainly the problem of determining an arc of curve, bounded by two given and having a smallest possible length. The problem of deter points mining and investigating a surface with given boundary and with a smallest possible area might then be considered as the most immediate two-dimensional variation problem. The classical work, concerned with the latter problem, is summed up in a beautiful and enthusiastic manner in DARBOUX'S Theorie generale des surfaces, vol. I, and in the first volume of the collected papers of H. A. SCHWARZ. The purpose of the present report is to give a picture of the progress achieved in this problem during the period beginning with the Thesis of LEBESGUE (1902). Our problem has always been considered as the outstanding example for the application of Analysis and Geometry to each other, and the recent work in the problem will certainly strengthen this opinion. It seems, in particular, that this recent work will be a source of inspiration to the Analyst interested in Calculus of Variations and to the Geometer interested in the theory of the area and in the theory of the conformal maps of general surfaces. These aspects of the subject will be especially emphasized in this report. The report consists of six Chapters. The first three Chapters are important tools or concerned with investigations which yielded either important ideas for the proofs of the existence theorems reviewed in the last three Chapters.
Winner of the American Library in Paris Book Award Named a Best Book of 2019 by BookPage During World War II, French villagers offered safe harbor to countless strangers—mostly children—as they fled for their lives. The same place offers refuge to migrants today. Why? In a remote pocket of Nazi-held France, ordinary people risked their lives to rescue many hundreds of strangers, mostly Jewish children. Was this a fluke of history, or something more? Anthropologist Maggie Paxson, certainties shaken by years of studying strife, arrives on the Plateau to explore this phenomenon: What are the traits that make a group choose selflessness? In this beautiful, wind-blown place, Paxson discovers a tradition of offering refuge that dates back centuries. But it is the story of a distant relative that provides the beacon for which she has been searching. Restless and idealistic, Daniel Trocmé had found a life of meaning and purpose—or it found him—sheltering a group of children on the Plateau, until the Holocaust came for him, too. Paxson's journey into past and present turns up new answers, new questions, and a renewed faith in the possibilities for us all, in an age when global conflict has set millions adrift. Riveting, multilayered, and intensely personal, The Plateau is a deeply inspiring journey into the central conundrum of our time.
This book is meant to give an account of recent developments in the theory of Plateau's problem for parametric minimal surfaces and surfaces of prescribed constant mean curvature ("H-surfaces") and its analytical framework. A comprehensive overview of the classical existence and regularity theory for disc-type minimal and H-surfaces is given and recent advances toward general structure theorems concerning the existence of multiple solutions are explored in full detail. The book focuses on the author's derivation of the Morse-inequalities and in particular the mountain-pass-lemma of Morse-Tompkins and Shiffman for minimal surfaces and the proof of the existence of large (unstable) H-surfaces (Rellich's conjecture) due to Brezis-Coron, Steffen, and the author. Many related results are covered as well. More than the geometric aspects of Plateau's problem (which have been exhaustively covered elsewhere), the author stresses the analytic side. The emphasis lies on the variational method. Originally published in 1989. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
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"This book recounts my journey through the Colorado Plateau, a journey through place and time and self.... During my explorations of more than three decades, I found a land that sears into my heart and soul, a place that has taught me and changed me. I also discovered a land of conflict and endurance, a land that has given birth to one of the great chapters in American history." --from the Introduction The Colorado Plateau, stretching across four states and covering nearly 80 million acres, is one of the most unique and spectacular landscapes in the world. Remote, rugged, and dry -- at once forlorn and glorious -- it is a separate place, a place with its own distinctive landscape, history, and future.In Fire on the Plateau, legal scholar and writer Charles Wilkinson relates the powerful story of how, over the past thirty years, he has been drawn ever more deeply into the redrock country and Indian societies of the Colorado Plateau. His work in the early 1970s as staff attorney for the newly formed Native American Rights Fund brought him into close contact with Navajo and Hopi people. His growing friendships with American Indians and increasing understanding of their cultures, along with his longstanding scholarship and experiences on federal public lands, led him to delve into the complicated history of the region.Wilkinson examines that history -- the sometimes violent conflicts between indigenous populations and more recent settlers, the political machinations by industry and the legal establishment, the contentious disputes over resources and land use -- and provides a compelling look at the epic events that have shaped the region. From centuries of habitation by native peoples to Mormon settlement, from the "Big Build-Up" of the post-World War II era to the increased environmental awareness of recent years, he explores the conquests of tribes and lands that have taken place, and the ways in which both have endured.Throughout, Wilkinson uses his own personal experiences as a lawyer working with Indian people and his heartfelt insights about a land that he grew to love to tie together the threads of the story. Fire on the Plateau is a vital and dynamic work that is sure to strike a chord with anyone interested in the past or future of the American Southwest.