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

Oliker, Vladimir

doi:10.1090/S0002-9939-2010-10439-2

A characterization of revolution quadrics by a system of partial differential equations
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by Vladimir I. Oliker PDF

Proc. Amer. Math. Soc. 138 (2010), 4075-4080 Request permission

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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