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

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

MathSciNet review:
1357407

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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.

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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