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International Atomic Energy Agency, Vienna, 1974 INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1974 GLOBAL ANALYSIS AND ITS APPLICATIONS Vol. II INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE GLOBAL ANALYSIS AND ITS APPLICATIONS LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE AT TRIESTE FROM 4 JULY TO 25 AUGUST 1972 ORGANIZED BY THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE In three volumes VOL. II INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1974 THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP) in Trieste was establishedby the International Atomic Energy Agency (IAEA) in 1964 under an agreement with the Italian Government, and with the a ssis­ tance of the City and University of Trieste. The IAEA and the United Nations Educational, Scientific and Cultural Organi­ zation (UNESCO) subsequently agreed to operate the Centre jointly from 1 January 1970. Member States of both organizations participate in the work of the Centre, the main purpose of which is to foster, through training and research, the advancement of theoretical physics, with special regard to the needs of developing countries. GLOBAL ANALYSIS AND ITS APPLICATIONS IAEA, VIENNA, 1974 STI/PUB/355 Printed by the IAEA in Austria September 1974 FOREWORD The International Centre for Theoretical Physics has maintained an interdisciplinary character in its research and training, programmes in different branches of theoretical physics. In pursuance of this objective, the Centre has organized extended research courses with a comprehensive and synoptic coverage in varying disciplines. The first of these — on plasma physics — was held in 1964; the second, in 1965, was concerned with the physics of particles. Between then and 1972, seven courses were organized; three on nuclear theory (during 1966, 1969 and 1971), three on physics of condensed matter (during 1967, 1970 and 1972), and one on Computing as a Language of Physics (1971). The Proceedings of all these courses have been published by the International Atomic Energy Agency. The present three volumes record the Proceedings of the tenth course held from 4 July to 25 August 1972 which dealt with Global Analysis and its Applications. Generous grants from the United Nations Development Pro­ gramme and from the Battelle Foundation are gratefully acknowledged. The programme of lectures was organized by Professors M. Dolcher (Trieste, Italy), J. Eells (Warwick, United Kingdom) and J.C . Zeeman (Warwick, United Kingdom). Abdus Salam EDITORIAL NOTE The papers and discussions incorporated in the proceedings published by the International Atomic Energy Agency are edited by the Agency's edi­ torial staff to the extent considered necessary for the reader's assistance. The views expressed and the general style adopted remain, however, the responsibility of the named authors or participants. For the sake of speed of publication the present Proceedings have been printed by composition typing and photo-offset lithography. Within the lim i­ tations imposed by this method, every effort has been made to maintain a high editorial standard; in particular, the units and symbols employed are to the fullest practicable extent those standardized or recommended by the competent international scientific bodies The affiliations of authors are those given at the time of nomination. The use in these Proceedings of particular designations of countries or territories does not imply any judgement by the Agency as to the legal status of such countries or territories, of their authorities and institutions or of the delimitation of their boundaries. The mention of specific companies or of their products or brand-names does not imply any endorsement or recommendation on the part of the International Atomic Energy Agency. CONTENTS OF VOL. II Geometric variational problems from a measure-theoretic point of view (IA E A -SM R -11/9) ..................................................................... 1 F .J . Almgren, J r . Area measures on a real vector space (IAEA-SMR-11 / 10) ................ 23 F . Brickell The differentiability of transformations which preserve geodesics (IA E A -SM R -11 / 11) ................................................................................................. 33 F . Brickell Stability theorems for R2-actions on manifolds (IAEA-SMR-11/12) .. 37 C. Camacho Introduction to minimal-surface theory (IAEA-SMR-11/13) ................ 43 E . D e Giorgi On complex varieties of nilpotent Lie algebras (after G. Favre) (IA E A -SM R -11/14) ................................................................................................. 47 P . D e l a Harpe On infinite-dimensional Lie groups acting on finite-dimensional m anifolds (IA E A -S M R -ll/1 5 ) ........................................................................... 59 P. D e l a Harpe Some properties of infinite-dimensional orthogonal groups (IA E A -SM R -11/16) ................................................................................................. 71 P. D e l a Harpe T heory of re sid u e s in se v e ra l v a ria b le s (I A E A - S M R -ll/17) ............... 79 P . Dolbeault On connections (IA E A -SM R -11 / 18) ................................................................... 97 P . Dolbeault Introduction to global calculus of variations (IAEA-SMR-11 / 19) ....... 113 H.I. E 1 i a s s о n Elementary survey of pseudo-differential operators and the wave-front set of a distribution (IAEA-SMR-11/20) ........................... 137 R .J . Elliott Boundary value problems for non-linear partial differential equations (IA E A -SM R -11/21 ) ........................................................................... 145 R .J. Elliott Gaussian measures on Banach spaces and manifolds (IA E A -SM R -11/22) ................................................................................................ 151 K.D . E 1 w o r t h у Sheaf cohomology, structures on manifolds and vanishing theory (IA E A -SM R -11/23) ................................................................................................ 167 M .J. Field . Complex analysis ori Banach spaces (IAEA-SMR-ll/24) ........................ 18Э M .J. Field Modern theory of billiards - an introduction (IAEA-SMR-11/25)........ 193 G. Gallavotti Some remarks on quasi-Abelian manifolds (IAEA-SMR-11/26) .......... 203 F . Gherardelli , A. Andreotti Life and death of the Bernstein problem (IAEA-SMR-11/27) ................ 207 E . G i u s t i Invariants of foliations (IAEA-SMR-11/28) ................................................... 215 C. Godbillon On the local solvability of linear partial differential equations (IA E A -SM R -11 /29) .................................................................................................. 221 H. Goldschmidt Compact operators and the minimax principle (IAEA-SMR-11/30) ... 225 R.A . Goldstein, R. Saeks R igidity and energy (IA E A -SM R -11 / 31) ........................................................ 233 R.A . Goldstein, P .J . Ryan Phase transitions in D-dimensional Ising lattices (IA E A -SM R -11/32) .................................................................................................. 245 R.A . Goldstein, J . J . Kozak Differential calculus in locally convex spaces (IAEA-SMR-11/33) ... 263 R.A . Graff Singularities in "soap bubbles" and "soap films" (IA E A -SM R -11 / 34) ................................................................................................. 271 Je an E . Taylor C au stics (IA EA -SM R -11/35) .................................................................................. 281 J . Guckenheimer The topological degree on Banach manifolds (IAEA-SMR-ll/36) ....... 291 C .A .S. Isnard Existence and non-existence for semi-linear elliptic equations (IA E A -SM R -11 /37) ................................................................................................. 315 J .L . K a z d a n An example of a strange three-dimensional surface in C2 (IA E A -SM R -11 /38) ................................................................................................. 323 J . J . Kohn A continuous change of topological type of Riemannian manifolds and its connection with the evolution of harmonic form s and spin stru c tu re s (IA E A -SM R -11/39) ............................................................. 329 J . Komorowski A new proof for regularity of solutions of elliptic differential o p e rato rs (IA E A -SM R -11/40) ........................................................................... 355 M. Kuranishi S e c re ta ria t of Sem in ar ............................................................................................ 363 lAEA-SMR-11/9 GEOMETRIC VARIATIONAL PROBLEMS FROM A MEASURE-THEORETIC POINT OF VIEW F.J. ALMGREN, Jr. * Department of Mathematics, Princeton University, Princeton, N.J., United States of America Abstract GEOMETRIC VARIATIONAL PROBLEMS FROM A MEASURE-THEORETIC POINT OF VIEW. Some phenomena of geometric variational problems are treated; in particular, a discussion of surfaces as measures, a regularity theorem, and estimates on singular sets are presented. "During the last three decades the subject of geometric measure theory has developed from a collection of isolated special results into a cohesive body of basic knowledge with an ample natural structure of its own, and with strong ties to many other parts of mathematics. These advances have given us deeper perception of the analytic and topological foundations of geometry, and have provided a new direction to the calculus of variations. Recently the methods of
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