The Geometry of Surfaces in Euclidean Spaces

The original version of this article was written more than five years ago with S.Z. Shefel’, a profound and original mathematician who died in 1984. Since then the geometry of surfaces has continued to be enriched with ideas and results. This has required changes and additions, but has not influenced the character of the article, the design of which originated with Shefel’. Without knowing to what extent Shefel’ would have approved the changes, I should nevertheless like to dedicate this article to his memory. (Yu.D. Burago)

[1]  A. V. Pogorelov.  Extrinsic geometry of convex surfaces , 1973 .

[2]  N. Kuiper,et al.  On C1-isometric imbeddings. II , 1955 .

[3]  J. H. Michael,et al.  Sobolev and mean‐value inequalities on generalized submanifolds of Rn , 1973 .

[4]  A. B. Borisov,et al.  Theory of dynamical affine and conformal symmetries as the theory of the gravitational field , 1974 .

[5]  Stability in Aleksandrov's problem for a convex body, one of whose projections is a ball , 1990 .

[6]  È R Rozendorn.  BOUNDED COMPLETE WEAKLY NONREGULAR SURFACES WITH NEGATIVE CURVATURE BOUNDED AWAY FROM ZERO , 1983 .

[7]  T. K. Milnor.  Efimov's theorem about complete immersed surfaces of negative curvature , 1972 .

[8]  Willy Feller,et al.  Krümmungseigenschaften Konvexer Flächen , 1936 .

[9]  Shing-Tung Yau,et al.  On the regularity of the monge‐ampère equation det (∂2 u/∂xi ∂xj) = f(x, u) , 1977 .

[10]  William K. Allard,et al.  On the first variation of a varifold , 1972 .

[11]  Ė. R. Rozendorn.  Surfaces of Negative Curvature , 1992 .

[12]  I. K. Sabitov.  Local Theory of Bendings of Surfaces , 1992 .

[13]  R. Osserman,et al.  The area of the generalized Gaussian image and the stability of minimal surfaces inSn and ℝn , 1982 .

[14]  J. Nash.  C 1 Isometric Imbeddings , 1954 .

[15]  T. Willmore.  Total curvature in Riemannian geometry , 1982 .

[16]  Ė. R. Rozendorn.  WEAKLY IRREGULAR SURFACES OF NEGATIVE CURVATURE , 1966 .

[17]  Changshou Lin.  The local isometric embedding in $\mathbf{R}^3$ of 2-dimensional Riemannian manifolds with nonnegative curvature , 1985 .

[18]  H. Fédérer.  Currents and area , 1961 .

[19]  H. Minkowski.  Volumen und Oberfläche , 1903 .

[20]  H. Lewy,et al.  On the non-vanishing of the Jacobian in certain one-to-one mappings , 1936 .

[21]  S. Z. Shefel'.  Two classes of k-dimensional surfaces in n-dimensional euclidean space , 1969 .

[22]  Lev Arkadʹevich Kaluzhnin,et al.  Die innere Geometrie der konvexen Flächen , 1955 .

[23]  Robert Connelly,et al.  A counterexample to the rigidity conjecture for polyhedra , 1977 .

[24]  Patrick J. Ryan,et al.  Tight and taut immersions of manifolds , 1985 .

[25]  A. Verner.  S. E. Cohn-Vossen's theorem on the total curvature of complete surfaces , 1968 .

[26]  Luther Pfahler Eisenhart,et al.  Introduction to Differential Geometry , 2015 .

[27]  L. Nirenberg.  The Weyl and Minkowski problems in differential geometry in the large , 1953 .

[28]  H. Minkowski.  Volumen und Oberfläche , 1903 .

[29]  N. V. Efimov.  SURFACES WITH A SLOWLY CHANGING NEGATIVE CURVATURE , 1966 .

[30]  A. Aleksandrov,et al.  Intrinsic geometry of convex surfaces , 1967 .

[31]  H. Jacobowitz.  Implicit Function Theorems and Isometric Embeddings , 1972 .

[32]  G. Herglotz.  Über die starrheit der eiflächen , 1943 .

[33]  V. Ogievetsky.  Infinite-dimensional algebra of general covariance group as the closure of finite-dimensional algebras of conformal and linear groups , 1973 .

[34]  I. Vekua.  Generalized Analytic Functions , 1962 .

[35]  M. Gromov,et al.  Partial Differential Relations , 1986 .

[36]  E G Poznyak,et al.  ISOMETRIC IMMERSIONS OF TWO-DIMENSIONAL RIEMANNIAN METRICS IN EUCLIDEAN SPACE , 1973 .

[37]  B. Kerékjártó,et al.  Vorlesungen über Topologie: I, Flächentopologie , 1923 .

[38]  N. Kuiper.  Minimal total absolute curvature for immersions , 1970 .

[39]  Erhard Heinz.  Über die Differentialungleichung 0 , 1959 .

[40]  W. J. Firey.  Christoffel's problem for general convex bodies , 1968 .

[41]  Bang-Yen Chen,et al.  Geometry of submanifolds , 1973 .

[42]  S. Yau,et al.  On the regularity of the solution of the n‐dimensional Minkowski problem , 1976 .

[43]  J. Hadamard,et al.  Les surfaces a courbures opposees et leurs lignes geodesique , 1898 .

[44]  A. Aleksandrov,et al.  Two-dimensional manifolds of bounded curvature , 1967 .

[45]  H. Lewy,et al.  On the Existence of a Closed Convex Surface Realizing a Given Riemannian Metric. , 1938, Proceedings of the National Academy of Sciences of the United States of America.

[46]  J. Nash.  The imbedding problem for Riemannian manifolds , 1956 .

[47]  A. B. Borisov,et al.  Theory of Dynamical Affine and Conformal Symmetries as Gravity Theory , 1974 .

[48]  Shiing-Shen Chern,et al.  On the Total Curvature of Immersed Manifolds , 1957 .

[49]  S. Klimentov.  On a method of constructing the solutions of boundary-value problems of the theory of bendings of surfaces of positive curvature , 1990 .

[50]  Bending of convex surfaces , 1980 .