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

TL;DR

This paper characterizes revolution quadrics via a specific nonlinear PDE system, linking solutions to geometric shapes like paraboloids and hyperboloids on Riemannian manifolds.

Contribution

It establishes a unique correspondence between solutions of a PDE system and classical rotationally symmetric hypersurfaces in Euclidean space.

Findings

Solutions imply the hypersurface is a paraboloid, ellipsoid, hyperboloid, or hyperplane.

The reciprocal of the PDE solution acts as the radial function of these shapes.

Topological and geometric conditions are derived from PDE existence.

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 solution its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in $\Rn$: paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.