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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-96-01650-9
PII: S 0002-9947(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