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The problem of embedding $ S\sp n$ into $ {\bf R}\sp {n+1}$ with prescribed Gauss curvature and its solution by variational methods


Author: V. I. Oliker
Journal: Trans. Amer. Math. Soc. 295 (1986), 291-303
MSC: Primary 53C45; Secondary 49F99, 58G30
MathSciNet review: 831200
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Abstract: A way to recover a closed convex hypersurface from its Gauss curvature is to find a positive function over $ {S^n}$ whose graph would represent the hypersurface in question. Then one is led to a nonlinear elliptic problem of Monge-Ampère type on $ {S^n}$. Usually, geometric problems involving operators of this type are too complicated to be suggestive for a natural functional whose critical points are candidates for solutions of such problems. It turns out that for the problem indicated in the title, such a functional exists and has interesting geometric properties. With the use of this functional, we obtain new existence results for hypersurfaces with prescribed curvature as well as strengthen some that are already known.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0831200-1
Article copyright: © Copyright 1986 American Mathematical Society