Embed Size (px)
Citation preview
Mathematical Surveys
and Monographs
Volume 130
Isometric Embedding of Riemannian Manifolds in Euclidean Spaces
Qing Han Jia-Xing Hong
American Mathematical Society
http://dx.doi.org/10.1090/surv/130
EDITORIAL COMMITTEE Jer ry L. Bona Pe te r S. Landweber Michael G. Eas twood Michael P. Loss
J. T . Stafford, Chair
2000 Mathematics Subject Classification. P r i m a r y 35J60, 53C21, 53C45, 58J05; Secondary 35J70, 35L45, 35L80, 35M10, 53C24.
For addi t ional information and upda te s on this book, visit w w w . a m s . o r g / b o o k p a g e s / s u r v - 1 3 0
Library of Congress Cataloging-in-Publicat ion D a t a Han, Qing.
Isometric embedding of Riemannian manifolds in Euclidean spaces / Qing Han, Jia-Xing Hong. p. cm. — (Mathematical surveys and monographs ; v. 130)
Includes bibliographical references and index. ISBN-13: 978-0-8218-4071-9 (alk. paper) ISBN-10: 0-8218-4071-1 (alk. paper) 1. Riemannian manifolds. 2. Isometrics (Mathematics). 3. Algebraic spaces. I. Hong, Jia-Xing,
1942—. II. Title. III. Series: Mathematical surveys and monographs ; no. 130.
QA649 .H345 2006 516.3/62—dc22 2006045898
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected].
© 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights
except those granted to the United States Government. Printed in the United States of America.
@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06
Dedicated to Shing-Tung Yau
To Yansu, Raymond and Thomas from Q. H.
Contents
Preface ix
A Brief History xi
Part 1. Isometric Embedding of Riemannian Manifolds 1
Chapter 1. Fundamental Theorems 3 1.1. Local Isometric Embedding of Analytic Metrics 4 1.2. Local Isometric Embedding of Smooth Metrics 9 1.3. Global Isometric Embedding of Smooth Metrics 16 Notes 31
Chapter 2. Surfaces in Low Dimensional Euclidean Spaces 33 2.1. Surfaces in R3 33 2.2. Isometric Immersions of Surfaces of Constant Curvature 38 2.3. Isometric Immersions of Surfaces in M4 40 Notes 42
Part 2. Local Isometric Embedding of Surfaces in R3 43
Chapter 3. Basic Equations 45 3.1. The Darboux Equation 45 3.2. The Gauss-Codazzi System 47 3.3. The Rozhdestvenskii-Poznyak System 51 Notes 53
Chapter 4. Nonzero Gauss Curvature 55 4.1. Positive Gauss Curvature 56 4.2. Negative Gauss Curvature 58 4.3. Appendix: Sobolev Spaces 63 4.4. Appendix: Some Lemmas 64 4.5. Appendix: Symmetric Hyperbolic Linear Differential Systems 68 Notes 70
Chapter 5. Gauss Curvature Changing Sign Cleanly 71 5.1. The Setting 71 5.2. Iterations 74 5.3. Symmetric Positive Linear Differential Systems 79 Notes 86
V l l l CONTENTS
Chapter 6. Nonnegative Gauss Curvature 6.1. The Setting 6.2. A Priori Estimates 6.3. Iterations Notes
Chapter 7. Nonpositive Gauss Curvature 7.1. The Setting 7.2. A Priori Estimates for the Linearized System 7.3. A Priori Estimates for the Cauchy Problem 7.4. Iterations 7.5. Remarks on the Hypothesis Notes
Part 3. Global Isometric Embedding of Surfaces in
Chapter 8. Deformation of Surfaces 8.1. The Rigidity of Surfaces 8.2. The Infinitesimal Rigidity of Surfaces 8.3. A Positive Disk of Not Infinitesimal Rigidity Notes
Chapter 9. The Weyl Problem 9.1. Geometric Preliminaries 9.2. Openness 9.3. Closedness Notes
Chapter 10. Complete Negatively Curved Surfaces 10.1. Nonexistence of Isometric Immersions 10.2. Existence of Isometric Immersions, Part I 10.3. Existence of Isometric Immersions, Part II Notes
Chapter 11. Boundary Value Problems 11.1. The Dirichlet Problem with Planar Boundaries 11.2. The Darboux System and Interior C2-Estimates 11.3. Global C2-Estimates Notes
Bibliography
Index
Preface
Given a smooth n-dimensional Riemannian manifold (M n ,g) , does it admit a smooth isometric embedding in Euclidean space R^ of some dimension N? This is a long-standing problem in differential geometry. When an isometric embedding in RN is possible for sufficiently large N, there arises a further question. What is the smallest possible value for JV? Those questions have more classical local versions in which solutions are sought only on a sufficiently small neighborhood of some specific point on the manifold.
In this book we present, in a systematic way, results concerning the isometric embedding of Riemannian manifolds in Euclidean spaces, both local and global, with the focus being on the isometric embedding of surfaces in R3. The book consists of three parts. In the first, we discuss some fundamental results of the isometric embedding of Riemannian manifolds in Euclidean spaces; these include the Janet-Cartan Theorem and Nash Embedding Theorem. In the second part, we study the local isometric embedding of surfaces in R3; we discuss metrics with Gauss curvature which is everywhere positive, negative, nonnegative, nonpositive, as well as the case of mixed sign. In the third part, we study the global isometric embedding of surfaces in R3; the main focus is on metrics on S2 with positive Gauss curvature and complete metrics in R2 with negative Gauss curvature. The emphasis of this book is on the PDE techniques for proving these results.
Differential geometers might, at first glance, consider the inclination toward analysis to be misplaced in these geometric problems and might even prefer less local coordinate calculations. However, all local calculations are designed to uncover the relevant PDE in the most efficient manner. The goal of this book is then to give a clean exposition of the techniques used in the analysis of these PDEs.
Completely omitted from the book is the local isometric embedding of higher-dimensional Riemannian manifolds in the Euclidean space of least dimension. Works on the higher-dimensional problems have involved much more differential geometry and methods such as exterior differential systems and are therefore far less accessible than the techniques presented in this book.
In integrating the results and techniques of a wide range of literature on the subject, we have tried to accommodate a broad readership as well as experts in the field. It is our objective that this book should provide a good entry into the area for second- or third-year graduate students. With this in mind, we have excluded everything that is technically complicated. Background knowledge is kept to an essential minimum. In Riemannian geometry, we assume only an acquaintance with basic concepts. In analysis, we assume the Cauchy-Kowalewsky theorem and some basic knowledge on elliptic and hyperbolic differential equations. On the
ix
x PREFACE
other hand, we hope that experts in the field will appreciate the organization of the results, covering the span of more than a century, into a unified whole.
Each chapter ends with bibliographical notes. Attributions are kept to a minimum in the body of the text, and the history and context of the works are expanded in the bibliographical notes.
All works quoted herein are already published.
Acknowledgments. We would like to express our gratitude to Shing-Tung Yau. Since the late 1970's, Yau has been promoting the problem of isometric embedding of Riemannian manifolds in Euclidean spaces. He taught a graduate course on this subject at the University of California at Berkeley in 1977 and at the National Tsing Hua University of Taiwan in 1989. His lecture notes were of interest among those studying the problem. It is fair to say that the research in this subject would not have reached its present level without his advocacy.
It is with pleasure we record here our thanks to our thesis advisors, Chaohao Gu (for J.-X. H.) and Fanghua Lin (for Q. H.). Our mathematical careers began with their patient guidance many years ago. Their persistent encouragement played an essential role in our research.
In preparing this book, we have benefitted greatly from discussions with many of our colleagues and friends. We especially thank Pengfei Guan, Yanyan Li and Chang-Shou Lin for many helpful discussions. A special thanks goes to Brian Smyth for reading the entire manuscript and for many suggested improvements.
Part 2 was presented by Q. H. in a series of seminars when he was a visitor at the Max-Planck Institute for Mathematics in Leipzig in 2002-2003. He would like to thank MPI for its hospitality. In particular, he would like to thank Stefan Muller for arranging his visit and providing him with a good opportunity to write this book.
Thanks also should be expressed to the American Mathematical Society, especially Edward Dunne and Deborah Smith, for their invaluable assistance in bringing this book to press.
The research related to this book was partially supported by a Sloan Foundation Fellowship and a grant from the National Science Foundation for Q. H. and a grant from the National Science Foundation of China for J.-X. H.
Qing Han, Jia-Xing Hong
A Brief History
Differential geometry began as the study of curves and surfaces in 3-space. The concept of a Riemannian manifold, an abstract manifold with a metric structure, was first formulated by Riemann in 1868 to generalize these classical objects. Naturally there arose the question of whether an abstract Riemannian manifold is simply a submanifold of some Euclidean space with its induced metric. This is the isometric embedding question. It has assumed a position of fundamental conceptual importance in differential geometry.
We briefly review four important aspects of the field of isometric embeddings of Riemannian manifolds in Euclidean space.
1. General Isometric Embedding of Riemannian Manifolds. In 1873, Schlaefli made the following conjecture: Every n-dimensional smooth
Riemannian manifold admits a smooth local isometric embedding in RSn, with sn — n(n + l ) /2 . It was more than 50 years later that an affirmative answer was given for the analytic case successively by Janet and Cartan; they proved in 1926-1927 that any analytic n-dimensional Riemannian manifold has a local analytic isometric embedding in RSn. Schlaefli's question for the smooth case when n = 2 was given renewed attention by Yau in the 1980's and 1990's.
For the global isometric embedding, Nash in 1954 and Kuiper in 1955 proved the existence of a global C1 isometric embedding of n-dimensional Riemannian manifolds in M2n+1 . For smooth isometric embeddings, the difficulty arises from the loss of derivatives in the attempt to solve the nonlinear equations corresponding to the isometric embedding. In an outstanding paper published in 1956, Nash introduced an important technique of using smoothing operators to make up for the loss of derivatives. He proved that any smooth n-dimensional Riemannian manifold admits a (global) smooth isometric embedding in the Euclidean space RN, for N = 3sn + 4n in the compact case and N = (n + l)(3sn + 4n) in the noncompact case. The technique proves to be extremely useful in solving nonlinear differential equations. It has been modified by many people, including Moser and Hormander, and is now known as the hard implicit function theorem, or Nash-Moser iteration.
Following Nash, one naturally looks for the smallest N. In his book Partial Differential Relations, published in 1986, Gromov studied various problems related to the isometric embedding of Riemannian manifolds. He proved that N = sn + 2n + 3 is enough for the compact case. Then in 1989, Giinther vastly simplified Nash's original proof. By rewriting the differential equations cleverly, he was able to employ the contraction mapping principle, instead of the Nash-Moser iteration, to construct solutions. Giinther also improved the dimension of the target space to
xi
xii A B R I E F HISTORY
N = max{sn + 2n, sn + n -f 5}. It is still not clear whether this is the best possible result on the dimension of the ambient space.
In 1970, Gromov and Rokhlin and Greene, independently, proved that any n-dimensional smooth Riemannian manifold admits a smooth isometric embedding in Rs™+n locally. The proof is based on the iteration scheme introduced by Nash.
For 2-dimensional Riemannian manifolds, better results are available. First, according to Gromov or Giinther, any compact 2-dimensional smooth Riemannian manifold can be isometrically embedded in R10 smoothly. When the manifolds have some special property, the dimension of the ambient space can be lowered. On the other hand, in 1973 Poznyak proved that any smooth 2-dimensional Riemannian manifold can be locally isometrically embedded in R4 smoothly. We are more interested in the question of whether we can isometrically embed a 2-dimensional Riemannian manifold in R3, locally or globally.
2. Local Isometric Embedding of Surfaces in R3. It was known to Darboux in 1894 that isometrically embedding a surface locally
in R3 is equivalent to finding a local solution of some nonlinear equation of the Monge-Ampere type. Such an equation is now called the Darboux equation; its type is determined by the sign of the Gauss curvature K. It is elliptic if K is positive and hyperbolic if K is negative. It is degenerate if K vanishes. Remarkably, even today, the local solvability of the Darboux equation in the general case is not covered by any known theory of partial differential equations.
The first attempt to establish the local isometric embedding of surfaces in R3
was not through the Darboux equation. In 1908, Levi proved the local isometric embedding in R3 of surfaces with negative curvature by using the equations of virtual asymptotes. It was several decades later that the Darboux equation attracted the attention of those interested in the isometric embedding. In the early 1950's, Hartman and Winter studied the Darboux equation in the case when the Gauss curvature K does not vanish and proved the existence of local solutions to the Darboux equation and hence the local isometric embedding in R3 in that case.
For a long time the case when K vanishes did not give way to the efforts of mathematicians. In 1985 and 1986, Lin made important breakthroughs. By a delicate analysis, he obtained the existence of sufficiently smooth local solutions of the Darboux equation and hence a sufficiently smooth isometric embedding in a neighborhood of p for the following two cases: K(p) = 0 and dK(p) ^ 0, or K > 0 in a neighborhood of the point p. Later, in 1987, Nakamura proved the existence of the smooth local isometric embedding if K(p) = 0, dK{p) — 0 and HessK(p) < 0. Evidently, K is nonpositive near the point p and the leading part of K is an irreducible quadratic polynomial. For the case of nonpositive Gauss curvature, Hong in 1991 also proved the existence of a sufficiently smooth local isometric embedding in a neighborhood of p if K — hg2rn, where h is a negative function and g is a function with g(p) = 0 and dg(p) ^ 0. In 2005, Han gave a simple proof of Lin's result that g admits a sufficiently smooth local isometric embedding in a neighborhood of p if K(p) = 0 and dK(p) ^ 0. All these results are based on a careful study of the Darboux equation.
In 2003, Han, Hong and Lin studied the local isometric embedding of surfaces in R3 by a different method. Instead of the Darboux equation, they studied a
A B R I E F HISTORY x m
quasilinear differential system equivalent to the Gauss-Codazzi system and proved the local isometric embedding for a large class of metrics with nonpositive Gauss curvature. They established the isometric embedding if some directional derivative of the Gauss curvature has a simple characterization for its zero set. This gives the results of Nakamura and Hong as special cases.
On the other hand, Pogorelov in 1972 constructed a C 2 , 1 metric g in B\ C R2
with a sign-changing Gauss curvature such that (Br,g) cannot be realized as a C2
surface in R3 for any r > 0. 3. Global Isometric Embedding of Surfaces in R3. In 1916, Weyl posed the following problem. Does every smooth metric on S2
with pointwise positive Gauss curvature admit a smooth isometric embedding in R3? The first attempt to solve the problem was made by Weyl himself. He used the continuity method and obtained a priori estimates up to the second derivatives. Twenty years later, Lewy solved the problem for an analytic metric g. In 1953, Nirenberg gave a complete solution under the very mild hypothesis that the metric g has continuous fourth derivatives. The result was extended to the case of continuous third derivatives of the metric by Heinz in 1962. In a completely different approach to the problem, Alexandroff in 1942 obtained a generalized solution of Weyl's problem as a limit of polyhedra. The regularity of this generalized solution was proved by Pogorelov in the late 1940's. In 1994 and 1995, Guan and Li, and Hong and Zuily independently generalized Nirenberg's result for metrics on S2 with nonnegative Gauss curvature.
The study of negatively curved surfaces in R3 is closely related to the interpretation of non-Euclidean geometry. The investigation of the isometric immersion of metrics with negative curvature goes back to Hilbert. He proved in 1901 that the full hyperbolic plane cannot be isometrically immersed in R3. A next natural step is to extend such a result to complete surfaces whose Gauss curvature is bounded above by a negative constant. The final solution of this problem was obtained by Efimov in 1963, more than sixty years later. Efimov proved that any complete negatively curved smooth surface does not admit a C2 isometric immersion in R3 if its Gauss curvature is bounded away from zero. Efimov's proof is very delicate and complicated. In the years following, Efimov extended his result in several ways.
Before the 1970's, the study of negatively curved surfaces was largely directed at nonexistence of isometric immersions in R3. As to existence, no result for complete negatively curved surfaces was known. In the 1980's, Yau proposed to find a sufficient condition for complete negatively curved surfaces to be isometrically immersed in R3. In 1993, Hong proved that complete negatively curved surfaces can be isometrically immersed in R3 if the Gauss curvature decays at a certain rate at infinity. His discussion was based on a differential system equivalent to the Gauss-Codazzi system.
Closely related to the global isometric embedding problem is the rigidity question. The first rigidity result was proved by Cohn-Vossen in 1927; this states that any two closed isometric analytic convex surfaces are congruent to each other. His proof was later considerably shortened by Zhitomirsky. In 1943, Herglotz gave a very short proof of the rigidity, assuming that the surfaces are three times continuously differentiable. Finally in 1962 it was extended by Sacksteder to surfaces with
XIV A B R I E F HISTORY
no more than two times continuously differentiable metrics. For compact surfaces with Gauss curvature of mixed sign, Alexandrov in 1938 introduced a class of surfaces satisfying some integral condition for its Gauss curvature and proved that any compact analytic surface with this condition is rigid. In 1963, Nirenberg generalized this result to smooth surfaces. To do this, he needed some extra conditions, one of which is not intrinsic.
4. Local Isometric Embedding of n-Dimensional Riemannian Manifolds in RSn. For the local isometric embedding of smooth n-dimensional Riemannian man
ifolds in RSri, the case n > 3 is sharply different from the case n = 2. For n = 2, there is only one curvature function, and it determines the type of the Darboux equation, that is, the equation for the isometric embedding of 2-dimensional Riemannian manifolds in R3. For n > 3, the role of curvature functions is not clear.
In 1983, Bryant, Griffiths and Yang studied the characteristic varieties associated with the differential systems for the isometric embedding in RSn of smooth n-dimensional Riemannian manifolds. They proved that these characteristic varieties are never empty if n > 3. This implies in particular that the differential systems for the isometric embedding in Rs™ of n-dimensional Riemannian manifolds are never elliptic for n > 3, no matter what assumptions are put on curvatures. This is a sharp difference from the case n = 2. A related result is the local rigidity proved by Berger, Bryant and Griffiths in 1983.
For n = 3, Bryant, Griffiths and Yang in 1983 studied the characteristic varieties in detail. They were able to classify the type of differential system for the isometric embedding by its curvature functions. Here an important quantity is the signature of the curvature tensor viewed as a symmetric linear operator acting on the space of 2-forms. They proved that any smooth 3-dimensional Riemannian manifold admits a smooth local isometric embedding in R6 if the signature is different from (0,0) and (0,1). Then in 1989, Nakamura and Maeda proved the existence of the smooth local isometric embedding in R6 of smooth 3-dimensional Riemannian manifolds if the curvature tensors are not zero. The key step in the proof is the local existence of solutions to differential systems of principal type.
248 11. BOUNDARY VALUE P R O B L E M S
Yau [230] posed two classes of boundary value problems for the isometric embedding of positive disks. The first class is a Dirichlet problem, which prescribes restrictions on the image of the boundary. The other is a Neumann problem, which prescribes the mean curvature of the image of the boundary. In fact, the general form of the Dirichlet problem was proposed earlier by Pogorelov [177]. This chapter is restricted to dealing with the Dirichlet problem. We refer the reader to [106] for the Neumann problem.
We should note that Gromov and Rohklin [71] constructed an analytic positive disc admitting no C2 isometric immersions in R3.
Bibliography
[1] Adams, R., Sobolev Spaces, Academic Press, New York, 1975. [2] Alexandroff, A. D., On a class of closed surfaces, Recueil Math. (Moscow), 4(1938), 69-77. [3] Alexandroff, A. D., Existence of a polyhedron and of a convex surface with given metric,
Dokl. Akad. NaukSSSR (N.S.), 50(1941), 103-106; Mat. Sbornik (N.S.), 11(53)(1942), 15-65. [4] Alexandroff, A. D., Intrinsic Geometry of Convex Surfaces, OGIZ, Moscow-Leningrad, 1948. [5] Amano, K., Global isometric embedding of a Riemannian 2-manifold with nonnegative cur
vature into a Euclidean 3-space, J. DifF. Geom., 34(1991), 49-83. [6] Aminov, Y. A., Isometric immersions of domains of n-dimensional Lobachevsky space in
(2n - 1)-dimensional Euclidean space, Mat. Sb., 111(1980), 402-433; Math. USSR-Sb., 39(1981), 359-386.
[7] Aminov, Y. A., Imbedding problems: Geometric and topological aspects, Problems in Geometry, vol. 13(Itogi Nauki i Tekhniki), VINITI, Moscow 1982, 119-157; J. Soviet Math., 25(1984), 1308-1331.
[8] Aminov, Y. A., Isometric immersions of domains of a three-dimensional Lobachevsky space in a five-dimensional Euclidean space, and motion of a rigid body, Mat. Sb., 122(1983), 12-30; Math. USSR-Sb., 50(1985), 11-30.
[9] Aminov, Y. A., Isometric immersions, with flat normal connection, of domains of n-dimensional Lobachevsky space into Euclidean spaces. A model of a gauge field, Mat. Sb., 137 (1988), 275-299; Math. USSR-Sb., 65(1990), 279-303.
[10] Aminov, Y. A., Geometry of the Grassmannian image of a local isometric immersion of an n-dimensional Lobachevsky space into a (2n — 1)-dimensional Euclidean space, Mat. Sb., 188 (1997), 3-28; Sb. Math., 188(1997), 1-27.
[11] Andrews, B., Notes on the isometric embedding problem and the Nash-Moser implicit function theorem, Surveys in analysis and operator theory (Canberra, 2001), 157-208, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002.
[12] Azov, D. G., Isometric embedding of n-dimensional metrics into Euclidean and spherical spaces, Vestnik Chelyabinsk. Univ. Ser. 3 Mat. Mekh., 1994, no. 1(2), 12-16.
[13] Bernstein, S., Uber ein geometrisches Theorem und seine Anwendung auf die partiellen Differential-gleichungen von elliptischen Typus, Math. Z., 26(1926), 551-558.
[14] Berger, E., Bryant, R., Griffiths, P., The Gauss equation and rigidity of isometric embed-dings, Duke Math. J., 50(1983), 803-892.
[15] Blanusa, D., Le plongement isometrique de la bande de Mobius infiniment large euclidienne dans un espace spherique, parabolique ou hyperbolique a quatre dimensions, Bull. Internat. Acad. Yougoslave. CI. Sci. Math. Phys. Tech. (N.S.), 12(1954), 19-23.
[16] Blanusa, D., Uber die Einbettung hyperbolischer Raume in euklidische Rdume, Monatsh. Math., 59(1955), 217-229.
[17] Blanusa, D., C°° -isometric imbeddings of cylinders with hyperbolic metric in Euclidean 7-space, Glas. Mat.-Fiz. i Astron., 11(1956), 243-246.
[18] Blanusa, D., A simple method of imbedding elliptic spaces in Euclidean spaces, Glas. Mat.-Fiz. i Astron., 12(1957), 190-200.
[19] Blanusa, D., Some problems of isometric embedding, Bull. Soc. Math. Phys. Serbie, 14(1962), 105-116.
[20] Blanusa, D., Embeddings of some 3-dimensional Euclidean spatial forms in spaces of constant curvature, Rad Jugoslav. Akad. Znan. Umjet., 331(1965), 237-262.
249
250 B I B L I O G R A P H Y
[21] Blaschke, W., Vorlesungen Uber Differentialgeometric, Vol. 1, Julis Springer, Berlin, 1924. [22] Blaschke, W., Uber eine geometrische Frage von Euklid bis heute, Hamburger Mathematis-
che Einzelschriften, 23, Teubner, Leipzig and Berlin, 1938. [23] Borisenko, A. A., Complete parabolic surfaces in Euclidean space, Ukrain. Geom. Sb., 28
(1985), 8-19; English transl., J. Soviet Math. 48 (1990), 6-14. [24] Borisenko, A. A., Isometric immersions of space forms into Riemannian and pseudo-
Riemannian spaces of constant curvature, Uspekhi Mat. Nauk, 56(2001), no. 3, 3-78; Russian Math. Surv., 56(2001), no. 3, 425-497.
[25] Bryant, R., Griffiths, P., Yang, D., Characteristics and existence of isometric embeddings, Duke Math. J., 50(1983), 893-994.
[26] Burago, Y. D., Shefel, S. Z., The geometry of surfaces in Euclidean spaces, Geometry, III, 1-85, Encyclopaedia Math. Sci., 48, Burago and Zalggaller (Eds.), Springer-Verlag, Berlin, 1992.
[27] Bushmelev, A. V., Isometric embeddings of an infinite flat Mobius strip and a flat Klein bottle in R4, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (1988), no. 3, 38-41; Mosc. Univ. Math. Bull., 43(1988), no. 3, 30-33.
[28] Caffarelli, L., Nirenberg, L., Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampere equation, Comm. Pure Appl. Math., 37(1984), 369-402.
[29] Caffarelli, L., Nirenberg, L., Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math., 155(1985), 261-301.
[30] Caffarelli, L., Cabre, X., Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ., Vol. 43, Amer. Math. Soc , Providence, RI, 1995.
[31] Calabi, E., Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geom., 1(1967), 111-125.
[32] Cartan, E., Sur la possibility de plonger un espace riemannian donne dans un espace eucli-dien, Ann. Soc. Pol. Math., 6(1927), 1-7.
[33] Chen, S.-X., Liu, L.-Q., Fourier integral operators of finite rank and their applications, I, II, (Chinese), Chin. Ann. Math., 6(A)(1985), 83-94, 539-550.
[34] Cheng, S. Y., Yau, S. T., On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math., 29(1976), 495-516.
[35] Chern, S. S., Some new characterizations of the Euclidean sphere, Duke Math. J., 12(1945), 279-290.
[36] Chern, S. S., Kuiper, N. H., Some theorems on the isometric embedding of compact Riemann manifolds in Euclidean space, Ann. of Math., 56(1952), 422-430.
[37] Chern, S. S., Lashof, R., On the total curvature of immersed manifolds, Amer. J. Math. 79(1957), 306-318; II, Michigan Math. J., 5(1958), 5-12.
[38] Cohn-Vossen, S. E., Zwei Sdtze uber die Starrheit der Eiflachen, Nach. Gesellschaft Wiss. Gottinger, Math. Phys. Kl.(1927), 125-134.
[39] Cohn-Vossen, S. E., Unstarre geochlossene Fldchen, Math. Annalen, 102(1930), 10-29. [40] Cohn-Vossen, S. E., The isometric deformability of surfaces in the large, Uspekhi Mat.
Nauk, 1(1936), 33-76. [41] Colombini, F., Spagnolo, S., An example of a weakly hyperbolic Cauchy problem not well
posed in C°°, Acta Math., 148(1982), 243-253. [42] Connelly, R., An immersed polyhedral surface which flexes, Indiana Univ. Math. J., 25(1976),
965-972. [43] Connelly, R., Conjectures and open questions in rigidity, Proceedings of the International
Congress of Mathematicians (Helsinki, 1978), 407-414, Acad. Sci. Fennica, Helsinki, 1980. [44] Connelly, R., Rigidity, Handbook of convex geometry, Vol. A, B, 223-271, Gruber P. M. and
Wills J. M. (Eds), North-Holland, Amsterdam, 1993. [45] Courant, R., DirichleVs Principle, Conformal Mapping and Minimal Surfaces, Interscience,
New York, 1950. [46] Darboux, G., Lecons sur la Theorie des Surfaces, Vol. 3, Gauthier-Villars, Paris, 1894. [47] Delanoe, Ph., Relations globalement regulieres de disques strictement convexes dans les
espaces d'Euclide et de Minkowski par la methode de Weingarten, Ann. Sci. Ec Norm. Super., IV, 21(1988), 637-652.
BIBLIOGRAPHY 251
[48] Ding, Q., Zhang, Y., Local isometric embeddings of surfaces into a 3-space, Chinese Ann. Math. Ser. B, 20(1999), no. 2, 215-222.
[49] do Carmo, M., and Dajczer, M., Local isometric immersions ofR2 into R4, J. Reine Angew. Math., 442(1993), 205-219.
[50] Dong G.-C., The semi-global isometric imbedding in R3 of two dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly, J. Partial Diff. Eqs., 6(1993), no.l , 62-79.
[51] Efimov, N. V., The impossibility in Euclidean 3-space of a complete regular surface with a negative upper bound of the Gaussian curvature, Dokl. Akad. Nauk SSSR (N.S.), 150(1963), 1206-1209; Soviet Math. Dokl., 4(1963), 843-846.
[52] Efimov, N. V., Generalization of singularities on surfaces of negative curvature, Mat. Sb., 64(1964), 286-320.
[53] Efimov, N. V., Surfaces with slowly varying negative curvature, Russian Math. Survey, 21(1966), 1-55.
[54] Efimov, N. V., A criterion of homeomorphism for some mappings and its applications to the theory of surfaces, Mat. Sb., 76(1968), 499-512.
[55] Efimov, N. V., Poznyak, E. G., A generalization of HilberVs theorem on surfaces of constant negative curvature, Dokl. Akad. Nauk SSSR (N.S.), 137(1961), 509-512; Soviet Math. Dokl., 4(1963), 843-846.
[56] Efimow, N. W., Flachenverbiegung im Grossen, Akademie-Verlag, Berlin, 1957. [57] Enomoto, K., Global properties of the Gauss image of flat surfaces in R4, Kodai Math. J.,
10(1987), 272-284. [58] Evans, L. C , Classical solutions of fully nonlinear, convex, second-order elliptic equations,
Comm. Pure Appl. Math., 35(1982), 333-363. [59] Friedman, A., Local isometric embedding of Riemannian manifolds with indefinite metrics,
J. Math. Mech., 10(1961), 625-649. [60] Friedrichs, K. O., Symmetric hyperbolic linear differential equations, Comm. Pure Appl.
Math., 7(1954), 345-392. [61] Friedrichs, K. O., Symmetric positive linear differential equations, Comm. Pure Appl. Math.,
11(1956), 333-418. [62] Garsia, A. M., An embedding of closed Riemann surfaces in Euclidean space, Comment.
Math. Helvetici, 35(1961), no. 2, 93-110. [63] Gilbarg, D., Trudinger, N., Elliptic Partial Differential Equations of Second Order, Second
Edition, Springer, Berlin, 1983. [64] Grauert, EL, On Levi's problem and the embedding of abstract real analytic manifolds, Ann.
of Math., 68(1958), 460-472. [65] Greene, R., Isometric embeddings of Riemannian and pseudo-Riemannian manifolds, Mem.
Amer. Math. Soc , No. 97, Amer. Math. Soc , Providence, RI, 1970. [66] Greene, R., Jacobowitz, H., Analytic isometric embeddings, Ann. of Math., 93(1971), 189-
204. [67] Greene, R., Wu, H., On the rigidity of punctured ovaloids, I, Ann. of Math., 94(1971), 1-20. [68] Greene, R., Wu, H., On the rigidity of punctured ovaloids, II, J. Diff. Geometry, 6(1972),
459-472. [69] Griffiths, P. A., Jensen, G. R., Differential systems and isometric embeddings, Princeton
Univ. Press, Princeton, NJ, 1987. [70] Gromov, M., Partial Differential Relations, Springer-Verlag, Berlin-Heidelberg, 1986. [71] Gromov, M. L., Rokhlin, V. A., Embeddings and immersions in Riemannian geometry,
Uspekhi Mat. Nauk., 25(1970), no. 5, 3-62; Russian Math. Survey, 25(1970), no. 5, 1-57. [72] Guan, B., Spruck, J., Boundary-value problems on Sn for surfaces of constant Gauss cur
vature, Ann. of Math., 138(1993), 601-624. [73] Guan, B., Guan, P., Hyper surf aces with prescribed curvatures, Ann. of Math., 156(2002),
655-674. [74] Guan, P., Li, Y., The Weyl problem with nonnegative Gauss curvature, J. Diff. Geometry,
39(1994), 331-342. [75] Giinther, M., Zum Einbettungssatz von J. Nash., Math. Nachr., 144(1989), 165-187.
252 BIBLIOGRAPHY
[76] Giinther, M., On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Ann. Global Anal. Geom., 7(1989), 69-77.
[77] Giinther, M., Isometric embeddings of Riemannian manifolds, Proceeding of the International Congress of Mathematicians, Kyoto, Japan, 1990, 1137-1143.
[78] Giinther, M., Zur lokalen Losbarleit nichtlinearer differentialgleichungen vom gemischten typ, Zeit. Analysis, 9(1990), 33-42.
[79] Hamiltion, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7(1982), 65-222.
[80] Han, Q., On the isometric embedding of surf aces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math., 58(2005), 285-295.
[81] Han, Q., Local isometric embedding of surfaces with Gauss curvature changing sign stably across a curve, Cal. Var. & P.D.E., 25(2005), 79-103.
[82] Han, Q., Hong, J.-X., Lin, C.-S., Local isometric embedding of surfaces with nonpositive gaussian curvature, J. Diff. Geometry, 63(2003), 475-520.
[83] Han, Q., Hong, J.-X., Lin, C.-S., Small divisors in nonlinear elliptic equations, Cal. Var. & P.D.E., 18(2003), 31-56.
[84] Han, Q., Hong, J.-X., Lin, C.-S., On Cauchy problems for degenerate hyperbolic equations, Trans. Amer. Math. Soc, 358(2006), 4021-4044.
[85] Hartman, P., On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvature, Trans. Amer. Math. Soc , 147(1970), 529-540.
[86] Hartman, P., Nirenberg, L., On spherical image maps whose Jacobians do not change sign, Amer. J. Math., 81(1959), 901-920.
[87] Hartman, P., Winter, P., On the embedding problem in differential geometry, Amer. J. Math., 72(1950), 553-564.
[88] Hartman, P., Winter, P., On the asymptotic curves of a surface, Amer. J. Math., 73(1951), 149-172.
[89] Hartman, P., Winter, P., Gaussian curvature and local embedding, Amer. J. Math., 73(1951), 876-884.
[90] Hartman, P., Winter, P., On hyperbolic partial differential equations, Amer. J. Math., 74(1952), 834-864.
[91] Heinz, E., On elliptic Monge-Ampere equation and Weyl's embedding problem, J. Anal. Math., 7(1959-1960), 1-52.
[92] Heinz, E., Neue a priori Abschatzungen fur den Ortsvektor einer Fldche positiver Gausscher Kriimmung durch ihr Linienelement, Math. Z., 74(1960), 129-157.
[93] Heinz, E., On Weyl's embedding problem, J. Math. Mech., 11(1962), 421-454. [94] Henke, W., Riemannsche Mdnnigfaltigkeiten konstanter positiver Kriimmung in euklidis-
chen Rdumen der Kodimension 2, Math. Ann., 193(1971), 265-278. [95] Henke, W, Isometrische Immersionen des n-dim. hyperbolischen Raumes H n in E 4 n _ 3 ,
Manuscripta Math., 34(1981), 265-278. [96] Herglotz, G., Uber die Starrheit der Eifldchen, Abh. Math. Sem. Hansischen Univ., 15(1943),
127-129. [97] Hilbert, D., Ueber fidchen von constanter Gausscher Kriimmung, Trans. Amer. Math. Soc ,
2(1901), 87-99. [98] Holmgren, E., Sur les surfaces a courbure constante negative, C.R. Acad. Sci. Paris,
134(1902), 740-743. [99] Hong, J.-X., Cauchy problem for degenerate hyperbolic Monge-Ampere equations, J. Partial
Diff. Equations, 4(1991), 1-18. [100] Hong, J.-X., Dirichlet problems for general Monge-Ampere equations, Math. Z., 209(1992),
289-306. [101] Hong, J.-X., Realization in R3 of complete Riemannian manifolds with negative curvature,
Comm. Anal. Geom., 1(1993), 487-514. [102] Hong, J.-X., Isometric embedding in M3 of complete noncompact nonnegatively curved sur
faces, Manuscripta Math., 94(1997), 271-286. [103] Hong, J.-X., Darboux equations and isometric embedding of Riemannian manifolds with
nonnegative curvature in R3, Chin. Ann. of Math., 2(1999), 123-136.
BIBLIOGRAPHY 253
[104] Hong, J.-X., Boundary value problems for isometric embedding in R3 of surfaces, Partial differential equations and their applications (Wuhan, 1999), 87-98, World Sci. Publishing, River Edge, New Jersey, 1999.
[105] Hong, J.-X., Recent developments of realization of surfaces in R3, First International Congress of Chinese Mathematicians (Beijing, 1998), 47-62, AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc , Providence, RI, 2001.
[106] Hong, J.-X., Positive disks with prescribed mean curvature on the boundary, Asian J. Math., 5(2001), 473-492.
[107] Hong, J.-X., Some new developments of realization of surfaces into R3, Proceedings of the International Congress of Mathematicians, Vol. Ill (Beijing, 2002), 155-165, Higher Ed. Press, Beijing, 2002.
[108] Hong, J.-X., Li, C , Infinitesimal nonrigidity of convex surfaces with planar boundary, Science in China, Ser. A Mathematics, 47(2004), no. 3, 1-9.
[109] Hong, J.-X., Zuily, C , Existence of C°° local solutions for the Monge-Ampere equation, Invent. Math., 89(1987), 645-661.
[110] Hong, J.-X., Zuily, C , Isometric embedding of the 2-sphere with nonnegative curvature in R3, Math. Z., 219(1995), 323-334.
[ I l l ] Hormander, L, The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62(1976), 1-52.
[112] Hormander, L, Implicit function theorems, Stanford Univ., Lecture Notes, 1977. [113] Hormander, L., On the Nash-Moser implicit function theorem, Ann. Acad. Sci. Fenn. Ser.
A. I. Math., 10(1985), 255-259. [114] Hormander, L., The Analysis of Linear Partial Differential Operators, I-IV, Second Edition,
Springer-Verlag, Berlin, 1990. [115] Hormander, L., Lectures on Nonlinear Hyperbolic Differential Equation, Springer, Berlin,
1996. [116] Iaia, J. A., Isometric embeddings of surfaces with nonnegative curvature in R3, Duke Math.
J., 67(1992), 423-459. [117] Ivanov, A. B., Isometric imbedding of a complete two-dimensional locally Euclidean mani
fold in a Euclidean space, Mat. Zametki, 13(1973), 427-429; Math. Notes, 13(1973), 258-259. [118] Iwasaki, N., Applications of Nash-Moser theory to nonlinear Cauchy problems, Proc. Sym-
pos. Pure Math., 45(1986) Part 1, 525-528. [119] Jacobowitz, H., Local isometric embeddings of surfaces into Euclidean four space, Indiana
Univ. Math. J., 21(1971/72), 249-254. [120] Jacobowitz, H., Implicit function theorems and isometric embeddings, Ann. of Math.,
95(1972), 191-225. [121] Jacobowitz, H., Local isometric embeddings, Seminar on Differential Geometry, 381-385,
Annals of Math. Studies, 102, Princeton University Press, 1982. [122] Jacobowitz, H., The global isometric embedding problem, Explorations in complex and Rie-
mannian geometry, 131-137, Contemp. Math., 332, Amer. Math. Soc , Providence, RI, 2003. [123] Janet, M., Sur la possibility de plonger un espace riemannian donne dans un espace eucli-
dien, Ann. Soc. Pol. Math., 5(1926), 38-43. [124] John, F., Partial Differential Equations, Fourth Edition, Springer-Verlag, New York, 1982. [125] Kaidasov, Zh., On regular isometric embedding in E3 of an expanding strip of the
Lobachevsky plane, Studies in the Theory of Surfaces in Riemannian Spaces, Leningrad. Gos. Ped. Inst., Leningrad, 1984, 119-129.
[126] Kaidasov, Zh., Shikin, E. V., On isometric embedding in E3 of a convex region of the Lobachevsky plane containing two horodisks, Mat. Zametki, 39(1986), 612-617; Math. Notes, 39(1986), 335-338.
[127] Kaidasov, Zh., Shikin, E. V., On isometric embedding in E 3 of a convex region of the Lobachevsky plane containing one horodisk, Vestnik Moskow Univ. Ser. I Mat. Mekh., 1986, no. 5, 79-81; Moscow Univ. Math. Bull., 41(1986).
[128] Kaidasov, Zh., Shikin, E. V., On isometric immersion in E 3 of expanding strips on L-type manifolds, Mat. Zametki, 42(1987), 842-853.
[129] Kajitani, K., Nishitani, T., The Hyperbolic Cauchy Problem, Lecture Notes in Math., 1505, Springer-Verlag, Berlin-Heidelberg-New York, 1991.
254 BIBLIOGRAPHY
[130] Kann, E., A new method for infinitesimal rigidity of surfaces with K > 0, J. Diff. Geometry, 4(1970), 5-12.
[131] Kann, E., An elementary proof of a finite rigidity problem by infinitesimal rigidity methods, Proc. Amer. Math. Soc , 60(1976), 252-258.
[132] Kann, E., Glidebending of general caps: an infinitesimal treatment, Proc. Amer. Math. Soc , 84(1982), 247-255.
[133] Kapouleas, N., Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math., 131(1990), 239-330.
[134] Kapouleas, N., Compact constant mean curvature surfaces in Euclidean three-space, J. Diff. Geom., 33(1991), 683-715.
[135] Kapouleas, N., Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math., 119(1995), 443-518.
[136] Kapouleas, N., Complete embedded minimal surfaces of finite total curvature, J. Diff. Geom., 47(1997), 95-169.
[137] Klingenberg, W., A Course in Differential Geometry, Graduate Texts in Mathematics, Vol. 51, Springer-Verlag, 1978.
[138] Klotz-Milnor, T., Efimov's theorem about complete immersed surfaces of negative curvature, Advances in Math., 8(1972), 474-543.
[139] Kohn, J. J., Nirenberg, L., Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math., 20(1967), 797-872.
[140] Kovaleva, G. A., An example of a surface which is homotopic to a tube and has a closed asymptotic line, Mat. Zametki, 3(1968), 403-413.
[141] Kovaleva, G. A., Absence of closed asymptotic lines on tubes of negative Gaussian curvature with a one-to-one spherical mapping, Fundam. Prikl. Mat., 1(1995), 953-977.
[142] Krylov, N. V., Boundedly nonhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk. SSSR Ser. Mat., 46(1982), 487-523; Math. USSR-Izv., 20(1983), 459-492.
[143] Krylov, N. V., Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk. SSSR Ser. Mat., 47(1983), 75-108; Math. USSR-Izv., 22(1984), 67-97.
[144] Kuiper, N. H., On C1-isometric embeddings, I, II, Indag. Math., 17(1955), 545-556, 683-689. [145] Kuiper, N. H., Isometric and short embeddings, Indag. Math., 21(1959), 11-25. [146] Lax, P. D., Phillips, R. S., Local boundary conditions for dissipative symmetric linear dif
ferential operators, Comm. Pure Appl. Math., 13(1960), 427-455. [147] Lebesgue, H., Sur le probleme de Dirichlet, Rend. Circ. Mat. Palermo, 24(1907), 371-402. [148] Levi, E., Sur Vapplication des equations integrales au probleme de Riemann, Nachr. Konigl.
Ges. Wiss. Gottingen Math.-Phys. Kl., (1908), 249-252. [149] Levine, J., Imbedding and immersion of real projective spaces, Proc. Amer. Math. Soc ,
14(1963), 801-803. [150] Lewy, H., A priori limitations for solutions of Monge-Ampere equations, I, Trans. Amer.
Math. Soc , 37(1935), 417-434. [151] Lewy, H., On the existence of a closed convex surface realizing a given Riemannian metric,
Proc. Nat. Acad. Sci., 24(1938), 104-106. [152] Lewy, H., On differential geometry in the large I (Minkowski's problem), Trans. Amer.
Math. Soc , 43(1938), 258-270. [153] Li, Y., Weinstein, G., A priori bounds for co-dimension one isometric embeddings, Amer.
J. Math., 121(1999), 945-965. [154] Lin, C.-S., The local isometric embedding in R3 of 2-dimensional Riemannian manifolds
with nonnegative curvature, J. Diff. Geometry, 21(1985), 213-230. [155] Lin, C.-S., The local isometric embedding in R3 of two dimensional Riemannian manifolds
with Gaussian curvature changing sign cleanly, Comm. Pure Appl. Math., 39(1986), 867-887.
[156] Mazzeo, R., Pacard, F., Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom., 9(2001), 169-237.
[157] Mazzeo, R., Pacard, F., Pollack, D., Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. Reine Angew. Math., 536(2001), 115-165.
[158] Minkowski, H., Volumen und Oberflache, Math. Annalen, 57(1903), 447-495.
BIBLIOGRAPHY 255
[159] Morrey, C , The analytic embedding of abstract real analytic manifolds, Ann. of Math., 68(1958), 159-201.
[160] Moser, J., A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U.S.A., 47(1961), 1824-1831.
[161] Moser, J., A rapidly converging iteration method and nonlinear partial differential equations I & II, Ann. Scuola Norm. Sup. Pisa, 20(1966), 265-313, 499-535.
[162] Nagumo, M., JJber das Anfangswertproblem Partieller Differentialgleichungen, Japan. J. Math., 18(1941), 41-47.
[163] Nakamura, G., Local isometric embedding of two-dimensional Riemannian manifolds into R3 with nonpositive Gaussian curvature, Tokyo J. Math., 10(1987), 241-257.
[164] Nakamura, G., Maeda, Y., Local smooth isometric embeddings of low dimensional Riemannian manifolds into Euclidean spaces, Tran. Amer. Math. Soc , 313(1989), 1-51.
[165] Nash, J., C1 isometric imbeddings, Ann. of Math., 60(1954), 383-396. [166] Nash, J., The embedding problem for Riemannian manifolds, Ann. of Math., 63(1956), 20-
63. [167] Nash, J., Analyticity of the solutions of implicit function problems with analytic data, Ann.
of Math., 84(1966), 345-355. [168] Nirenberg, L., The Weyl and Minkowski problems in differential geometry in the large,
Comm. Pure Appl. Math., 6(1953), 337-394. [169] Nirenberg, L., On nonlinear elliptic partial differential equations and Holder continuity,
Comm. Pure Appl. Math., 6(1953), 103-156. [170] Nirenberg, L., Rigidity of a class of closed surfaces, Nonlinear Problems, edited by R. Lang,
177-193, University of Wisconsin Press, 1963. [171] Nirenberg, L., An abstract form of the nonlinear Cauchy-Kowalewski theorem, J. Diff.
Geometry, 6(1972), 561-576. [172] Pogorelov, A. V., Intrinsic estimates for the derivatives of the radius vector of a point on
a regular convex surface, Dokl. Akad. Nauk SSSR (N.S.), 66(1949), 805-808. [173] Pogorelov, A. V., On the regularity of convex surfaces with regular metric, Dokl. Akad.
Nauk SSSR (N.S.), 66(1949), 1051-1053. [174] Pogorelov, A. V., On convex surfaces with regular metric, Dokl. Akad. Nauk SSSR (N.S.),
67(1949), 791-794. [175] Pogorelov, A. V., Regularity of a convex surface with given Gaussian curvature (Russian),
Mat. Sbornik (N.S.), 31(1952), 88-103. [176] Pogorelov, A. V., An example of a two-dimensional Riemannian metric that does not admit
a local realization in E3 , Dokl. Akad. Nauk SSSR (N.S.), 198(1971) 42-43; Soviet Math. Dokl., 12(1971), 729-730.
[177] Pogorelov, A. V., Extrinsic Geometry of Convex Surfaces, Amer. Math. Soc , Providence, RI, 1973.
[178] Pogorelov, A. V., The Minkowski Multidimensional Problem, John Wiley, New York, 1978. [179] Poznyak, E. G., On regular global realization of two-dimensional metrics of negative cur
vature, Dokl. Akad. Nauk SSSR (N.S.), 170(1966), 786-789; Soviet Math. Dokl., 7(1966), 1288-1291.
[180] Poznyak, E. G., Realization in the large of two-dimensional analytic metrics of sign-changing curvature, Ukrain Geometr. Sb., 7(1969), 89-97.
[181] Poznyak, E. G., Isometric embeddings of two-dimensional Riemannian metrics in Euclidean space, Uspekhi Mat. Nauk., 28(1973), no. 4, 47-76; Russian Math. Survey, 28(1973), no. 4, 47-77.
[182] Poznyak, E. G., Isometric embedding in E 3 of some noncompact parts of the Lobachevsky plane, Mat. Sb., 102(1977), 3-12.
[183] Poznyak, E. G., Shikin E. V., Analytic tools of the theory of imbeddings of two-dimensional manifolds of negative curvature, Izv. Vyssh. Uchebn. Zaved. Mat., (1986), 56-60.
[184] Poznyak, E. G., Shikin E. V., Small parameters in the theory of isometric embeddings of two-dimensional Riemannian manifolds in Euclidean spaces, Amer. Math. Soc. Transl. (2), 176(1996), 151-192.
256 BIBLIOGRAPHY
[185] Poznyak, E. G., Sokolov, D. D., Isometric immersions of Riemannian spaces in Euclidean spaces, Algebra, Geometry, and Topology, vol. 15 (Itogi Nauki i Tekhniki), VINITI, Moscow, 1977, 173-211; J. Soviet Math., 14(1980), 1407-1428.
[186] P. Rabinowitz, A rapid convergence method and a singular perturbation problem, Ann. Inst. Henri Poineare, 1(1984), 1-17.
[187] Rembs, E., Zur Verbiegung von Flachen im Grossen, Math. Z., 56(1952), 271-279. [188] Rozendorn, E. R., Realization of the metric ds2 = du2-\-f(u)dv2 in five-dimensional Euclid
ean space, Akad. Nauk Armyan SSR. Dokl., 30(1960), 197-199. [189] Rozendorn, E. R., On complete surfaces of negative curvature K < — 1 in the Euclidean
spaces E 3 and E4 , Mat. Sb., 58(1962), 453-478. [190] Rozendorn, E. R., Weakly irregular surfaces of negative curvature, Uspekhi Mat. Nauk.,
21(1966), no. 5, 59-116; Russian Math. Surveys, 21(1966), no. 5, 57-112. [191] Rozendorn, E. R., On the immersion of two-dimensional Riemannian metrics into four-
dimensional Euclidean space, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1979, no. 2, 47-50; Moscow Univ. Math. Bull., 34(1979), no. 2, 49-52.
[192] Rozendorn, E. R., Surfaces of negative curvature, Geometry, III, 87-178, Encyclopaedia Math. Sci., 48, Burago and Zalggaller (Eds.), Springer-Verlag, Berlin, 1992.
[193] Rozhdestvenskh, B., Quasilinear hyperbolic systems for theory of surfaces, Dokl. Akad. Nauk. SSSR (N.S.), 143(1962), 50-52; Soviet Math. Dokl., 3(1962), 351-353.
[194] Sabitov, I. K., Isometric immersions and embeddings of locally Euclidean metrics in M3, Sibirsk. Mat. Zh., 26(1985), no. 3, 156-167; Siberian Math. J., 26 (1985), 431-440.
[195] Sabitov, I. K., Isometric immersions of the Lobachevsky plane in E4 , Sibirsk. Mat. Zh., 30(1989), no. 5, 179-186; Siberian Math. J., 30 (1989), 805-811.
[196] Sabitov, I. K., Local theory of bendings of surfaces, Geometry, III, 179-256, Encyclopaedia Math. Sci., 48, Burago and Zalggaller (Eds.), Springer-Verlag, Berlin, 1992.
[197] Sacksteder, R., The rigidity of hyper surf aces, J. Math. Mech., 11(1962), 929-939. [198] Sa Erp, R., Tubian, E., Discrete and nondiscrete isometric deformations of surfaces in R3,
Sibirsk. Mat. Zh. 43(2002), no. 4, 887-893; Siberian Math. J., 43(2002), no. 4, 714-718. [199] Schlaefli, L., Nota alia memoria del Sig. Beltrami, Sugli spazii di curvatura constante, Ann.
di mat., 2e serie, 5(1871-1873), 170-193. [200] Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press,
New York, 1993. [201] Schwartz, J., On Nash's implicit functional theorem, Comm. Pure Appl. Math., 13(1960),
509-530. [202] Schwartz, J., Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences,
New York, 1963-1964. [203] Sergeraert, F., Une generalisation du theoreme des fonctions implicites de Nash, C. R.
Acad. Sci. Paris Ser. A, 270(1970), 861-863. [204] Shefel, S. Z., Totally regular isometric immersions in Euclidean space, Sibirsk. Mat. Zh.,
11(1970), 442-460; Siberian Math. J., 11(1985), 337-350. [205] Shikin, E. V., On the existence of solutions of the Peters on-Codazzi-Gauss system of equa
tions, Mat. Zametki, 17(1975), 765-781. [206] Shikin, E. V., Isometric embedding in E 3 of noncompact regions with nonpositive curvature,
Mat. Zametki, 25(1979), 785-797. [207] Shikin, E. V., On isometric embedding of two-dimensional manifolds with negative curvature
by the Darboux method, Mat. Zametki, 27(1980), 779-794. [208] Shikin, E. V., On the equations of isometric immersions of two-dimensional manifolds of
negative curvature in a three-dimensional Euclidean space, Mat. Zametki, 31(1982), 601-612.
[209] Shikin, E. V., The isometric embedding problem and Monge-Ampere equations of hyperbolic type, Trudy Inst. Mat. (Novosibirsk), 14(1989), Sovrem. Probl. Geom. Analiz., 245-258.
[210] Smyth, B., Xavier, F., Efimov's theorem in dimension greater than two, Invent. Math., 90(1987), 443-450.
[211] Spivak, M., A Comprehensive Introduction to Differential Geometry, Vol. I-V, Publish or Perish, Inc., Boston, MA, 1972-1975.
BIBLIOGRAPHY 257
[212] Struwe, M., Plateau's Problem and the Calculus of Variations, Math. Notes 35, Princeton University Press, Princeton, NJ, 1989. Thompson, A. C , Minkowski Geometry, Cambridge University Press, New York, 1996. Tompkins, C , Isometric embedding of flat manifolds in Euclidean space, Duke Math. J., 5(1939), 58-61. Tompkins, C , A flat Klein bottle isometrically embedded in Euclidean J^-space, Bull. Amer. Math. Soc, 47(1941), 508. Tunitskii, D. V., On regular isometric embedding in E 3 of unbounded regions with negative curvature, Mat. Sb., 134(1987), 119-134; Math. USSR Sb., 62(1989), 121-138. Tunitskii, D. V., Global regular isometric embedding of two-dimensional metrics with non-positive curvature, Mat. Sb., 183(1992), 65-80; Russian Acad. Sci. Sb. Math., 76(1993), 317-329. Vinogradskii, A. S., Boundary properties of surfaces with slowly changing negative curvature, Mat. Sb., 82(1970), 285-299; USSR. Math.-Sb., 11(1970), 257-271. Vorobeva, L. I., On impossibility of C2-isometric immersion of the Lobachevsky half-plane into E3 , Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1975, no. 5, 42-46; Moscow Univ. Math. Bull., 30(1976), no. 5/6, 32-35. Wente, H., Counterexample to a conjecture of H. Hopf, Pacific J. Math., 121(1986), 193-243. Wente, H., Constant mean curvature immersions of Enneper type, Mem. Amer. Math. Soc , 100(1992), no. 478. Weyl, H., Uber die Bestimmheit einer geschlossenen konvex Flache durch ihr Linienelement, Vierteljahresschrift der nat.-Forsch. Ges. Zurich, 61(1916), 40-72. Whitney, H., Differentiable manifolds, Ann. of Math., 31(1936), 645-680. Whitney, H., On the topology of differentiable manifolds, Lectures in Topology, Univ. of Michigan Press, 1941. Whitt , L., Isometric homotopy and codimension-two isometric immersions of the n-sphere into Euclidean space, J. Diff. Geom., 14(1979), 295-302. Xavier, F., A non-immersion theorem for hyperbolic manifolds, Comment. Math. Helv., 60(1985), 280-283. Yagdjian, K., The Cauchy Problem for Hyperbolic Operators: Multiple Characteristics. Micro-Local Approach, Math. Top., 12, Akademie Verlag, Berlin, 1997. Yanenko, N. N., Some questions of the theory of embeddings of Riemannian metrics in Euclidean spaces, Uspekhi Mat. Nauk., 8(1953), no. 1, 21-100. Yau, S.-T., Problem Section, Seminar on Differential Geometry, 669-706, Annals of Math. Studies, 102, Princeton University Press, Princeton, NJ, 1982. Yau, S.-T., Open problems in geometry, Chern - A Great Geometer of the Twentieth Century, 275-319, International Press, Hong Kong, 1992. Yau, S.-T., Review of geometry and analysis, Asian J. Math., 4(2000), 235-278. Zehnder, E., Generalized implicit function theorems with applications to some small divisor problems I & II, Comm. Pure Appl. Math., (28)1975, 91-141; (29)1976, 49-113.
[233] Zhitomirsky, O. K., Sur la non-flexibilite des ovaloides, Dokl. Akad. Nauk. SSSR (N.S.), 25(1939), 347-349.
Index
admissible boundary conditions, 80 Alexandroff, A. D., 165 asymptotic curves, 148 asymptotic quadrangles, 192
Cartan, E., 31 Cauchy-Kowalevski Theorem, 7 characteristic quadrangles, 192 Christoffel symbols, 34 Codazzi equations, 34 Cohn-Vossen Theorem, 145, 165 conjugate isothermal coordinates, 229 convex caps, 153 Courant-Lebesgue Lemma, 231 curvature tensor, 34
Darboux equations, 46, 168 Darboux systems, 230
Efimov, N. V., 224
flat Klein bottle, 38 flat metrics, 37 flat Mobius band, 39 flat torus, 38 free maps, 5 Friedrichs, K. O., 86 fundamental forms
the first fundamental form, 33 the second fundamental form, 33
Fundamental Theorem of Surfaces, 35
Gauss curvature, 35 Gauss equation, 34 Gauss-Bonnet formula, 196 Gauss-Codazzi system, 34 geodesic coordinates, 6, 38 geodesic curvature, 37 geodesic polar coordinates, 195 Greene, R., 31 Gromov, M. L., 31, 248 Gunther, M., 31, 86
Han, Q., 53, 86, 141
Harnack inequality, 236 Hartman, P., 70 Heinz, E., 247 Herglotz integral formula, 147 Hong, J.-X., 53, 141, 224, 247 hyperbolic plane, 39
infinitesimal isometric deformations, 154 infinitesimal rigidity, 154 interpolation inequalities, 63 isometric deformation, 154 isothermal coordinates, 232
Janet, M., 31 Janet system, 6 Janet-Cartan Theorem, 4
Kohn, J. J., 107
Levi, E., 70 Lin, C.-S., 53, 86, 107, 141
Maeda, Y., 86, 141 mean curvature, 35 method of continuity, 167, 246 Minkowski functions, 169, 241
Nakamura, G., 86, 141 Nash Embedding Theorem, 4 Nash, J., 31, 107 Nash-Moser iterations, 96, 130 Nirenberg, L., 107, 165, 188 normal coordinate systems, 71 normal curvatures, 149
osculating spaces, 4
Pogorelov, A. V., 188 Poincare-Bendixson Theorem, 151 positive disks, 225 Poznyak, E. G., 42 projective plane, 39 proper domains, 192
rigidity, 145
259
260 INDEX
Rodrigues formula, 149 Rokhlin, V. A., 31, 248 rotation vectors, 155 Rozhdestvenskii-Poznyak systems, 52
Sergeraert, F., 141 smoothing operators, 97 Sobolev embedding, 63 Sobolev functions, 63 symmetric hyperbolic differential systems,
69 symmetric positive differential systems, 79
torus-like surfaces, 148 Tricomi equations, 86
uniformization theorems, 170, 246
Weingarten equations, 34 Weyl problem, 167 Whitney, H., 31 Winter, P., 70
Yau, S.-T., 42, 224, 248
Titles in This Series
130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, 2006
129 Wil l iam M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its , Andrei A. Kapaev, and Victor Yu.
Novokshenov, Painleve transcendents, 2006 127 Nikolai Chernov and R o b e r t o Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson , and Wil l iam T. Ross , The Cauchy Transform,
2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor if-Theory, 2006 123 Barbara Fantechi, Lothar Gottsche , Luc Illusie, S teven L. Kle iman, Ni t in
Nitsure , and Ange lo Vistol i , Fundamental algebraic geometry: Grothendieck's FGA explained, 2005
122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005
121 Anton Zettl , Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian M a and Shouhong Wang, Geometric theory of incompressible flows with
applications to fluid dynamics, 2005 118 Alexandru B u i u m , Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 Wil l iam G. Dwyer , Phi l ip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith ,
Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D . Smith , The classification of quasithin groups
II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D . Smith , The classification of quasithin groups I.
Structure of strongly quasithin if-groups, 2004 110 Benne t t Chow and D a n Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups,
2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D . Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with
applications to economics, second edition, 2003 104 Graham Everest , Alf van der Poorten , Igor Shparlinski, and Thomas Ward,
Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre,
Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Phi l ip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guil lemin, Viktor Ginzburg, and Yael Karshon, Moment maps,
cobordisms, and Hamiltonian group actions, 2002
TITLES IN THIS SERIES
97 V . A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Mart in Markl, Steve Shnider, and J im Stasheff, Operads in algebra, topology and
physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D . Neuse l and Larry Smith , Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2:
Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1:
Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery , A tour of subriemannian geometries, their geodesies and
applications, 2002 90 Christ ian Gerard and Izabella Laba, Multiparticle quantum scattering in constant
magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Prenkel and David Ben-Zvi , Vertex algebras and algebraic curves, second
edition, 2004 87 Bruno Poizat , Stable groups, 2001 86 Stanley N . Burris, Number theoretic density and logical limit laws, 2001 85 V . A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with
corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant
differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential
operators, and layer potentials, 2000 80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module
theory, 2000 79 Joseph A. Cima and Wil l iam T. Ross , The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt , KP or mKP: Noncommutative mathematics of Lagrangian,
Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz , The semicircle law, free random variables and entropy,
2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W . Stroock, An introduction to the analysis of paths on a Riemannian manifold,
2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems
and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor) , Surgery on compact manifolds, second edition,
1999 68 David A. Cox and Sheldon Katz , Mirror symmetry and algebraic geometry, 1999
For a complete list of t i t les in this series, visit t he AMS Bookstore at w w w . a m s . o r g / b o o k s t o r e / .
