Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature

Wang, Xu-Jia

doi:10.1090/S0002-9947-96-01650-9

Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature
HTML articles powered by AMS MathViewer

by Xu-Jia Wang PDF

Trans. Amer. Math. Soc. 348 (1996), 4501-4524 Request permission

Abstract:

Let $f(x)$ be a given positive function in $R^{n+1}$. In this paper we consider the existence of convex, closed hypersurfaces $X$ so that its Gauss-Kronecker curvature at $x\in X$ is equal to $f(x)$. This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389–410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the existence of unstable solutions to the problem.

References

Ju. D. Burago and V. A. Zalgaller, Geometricheskie neravenstva, “Nauka” Leningrad. Otdel., Leningrad, 1980 (Russian). MR 602952
Kung-ching Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1196690, DOI 10.1007/978-1-4612-0385-8
Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the solution of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. MR 423267, DOI 10.1002/cpa.3160290504
K. S. Chou and X. J. Wang, The logarithmic Gauss curvature flow, preprint.
Ph. Delanoë, Plongements radiaux $S^n\hookrightarrow \textbf {R}^{n+1}$ à courbure de Gauss positive prescrite, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 635–649 (French, with English summary). MR 839688, DOI 10.24033/asens.1498
V. I. Oliker, Hypersurfaces in $\textbf {R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampère type, Comm. Partial Differential Equations 9 (1984), no. 8, 807–838. MR 748368, DOI 10.1080/03605308408820348
V. I. Oliker, The problem of embedding $S^n$ into $\textbf {R}^{n+1}$ with prescribed Gauss curvature and its solution by variational methods, Trans. Amer. Math. Soc. 295 (1986), no. 1, 291–303. MR 831200, DOI 10.1090/S0002-9947-1986-0831200-1
Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg. MR 0478079
Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465–477. MR 341351
Kaising Tso, Convex hypersurfaces with prescribed Gauss-Kronecker curvature, J. Differential Geom. 34 (1991), no. 2, 389–410. MR 1131436
Kaising Tso, On a real Monge-Ampère functional, Invent. Math. 101 (1990), no. 2, 425–448. MR 1062970, DOI 10.1007/BF01231510
D. G. Azov, Imbedding by the Blanuša method of certain classes of complete $n$-dimensional Riemannian metrics in Euclidean spaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1985), 72–74, 98 (Russian). MR 814279
Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR 645762