Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature
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- 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
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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
- 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.
- © Copyright 1996 American Mathematical Society
- 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