Existence of convex hypersurfaces

with prescribed Gauss-Kronecker curvature

Author:
Xu-Jia Wang

Journal:
Trans. Amer. Math. Soc. **348** (1996), 4501-4524

MSC (1991):
Primary 53C45, 58G11, 35J60

MathSciNet review:
1357407

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a given positive function in . In this paper we consider the existence of convex, closed hypersurfaces so that its Gauss-Kronecker curvature at is equal to . 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.

**1.**Ju. D. Burago and V. A. Zalgaller,*Geometricheskie neravenstva*, “Nauka” Leningrad. Otdel., Leningrad, 1980 (Russian). MR**602952****2.**Kung-ching Chang,*Infinite-dimensional Morse theory and multiple solution problems*, Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR**1196690****3.**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**0423267****4.**K. S. Chou and X. J. Wang,*The logarithmic Gauss curvature flow*, preprint.**5.**Ph. Delanoë,*Plongements radiaux 𝑆ⁿ↪𝑅ⁿ⁺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****6.**V. I. Oliker,*Hypersurfaces in 𝑅ⁿ⁺¹ with prescribed Gaussian curvature and related equations of Monge-Ampère type*, Comm. Partial Differential Equations**9**(1984), no. 8, 807–838. MR**748368**, 10.1080/03605308408820348**7.**V. I. Oliker,*The problem of embedding 𝑆ⁿ into 𝑅ⁿ⁺¹ with prescribed Gauss curvature and its solution by variational methods*, Trans. Amer. Math. Soc.**295**(1986), no. 1, 291–303. MR**831200**, 10.1090/S0002-9947-1986-0831200-1**8.**Aleksey Vasil′yevich Pogorelov,*The Minkowski multidimensional problem*, 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; Scripta Series in Mathematics. MR**0478079****9.**Robert C. Reilly,*Variational properties of functions of the mean curvatures for hypersurfaces in space forms*, J. Differential Geometry**8**(1973), 465–477. MR**0341351****10.**Kaising Tso,*Convex hypersurfaces with prescribed Gauss-Kronecker curvature*, J. Differential Geom.**34**(1991), no. 2, 389–410. MR**1131436****11.**Kaising Tso,*On a real Monge-Ampère functional*, Invent. Math.**101**(1990), no. 2, 425–448. MR**1062970**, 10.1007/BF01231510**12.**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****13.**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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
53C45,
58G11,
35J60

Retrieve articles in all journals with MSC (1991): 53C45, 58G11, 35J60

Additional Information

**Xu-Jia Wang**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China

Address at time of publication:
School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

Email:
wang@pell.anu.edu.au

DOI:
https://doi.org/10.1090/S0002-9947-96-01650-9

Keywords:
Gauss curvature,
convex hypersurface,
existence

Received by editor(s):
April 3, 1995

Received by editor(s) in revised form:
July 5, 1995

Additional Notes:
This work was finished while the author was visiting the Mathematical Section of the International Center for Theoretical Physics. He would like to thank the center for its support.

Article copyright:
© Copyright 1996
American Mathematical Society