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A characterization of revolution quadrics by a system of partial differential equations

Author: Vladimir I. Oliker
Journal: Proc. Amer. Math. Soc. 138 (2010), 4075-4080
MSC (2010): Primary 53A05, 53C40
Published electronically: June 29, 2010
MathSciNet review: 2679628
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Abstract: It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such a solution, its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in ${\mathbb R}^{n+1}$: paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.

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Additional Information

Vladimir I. Oliker
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

Keywords: Hypersurfaces of revolution, Obata’s theorem
Received by editor(s): June 6, 2008
Received by editor(s) in revised form: February 20, 2009
Published electronically: June 29, 2010
Additional Notes: The research of the author was partially supported by National Science Foundation grant DMS-04-05622.
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society