On compact hypersurfaces in a Riemannian vector bundle with prescribed vertical Gaussian curvature

TL;DR

This paper constructs smooth hypersurfaces in a Riemannian vector bundle with a prescribed vertical Gaussian curvature by solving a nonlinear Monge-Ampère type PDE, ensuring regularity through a radial graph approach.

Contribution

It introduces a novel method to ensure smooth solutions for hypersurfaces with prescribed vertical Gaussian curvature in Riemannian vector bundles using a radial graph formulation.

Findings

Existence of smooth hypersurfaces with prescribed vertical Gaussian curvature.

Solution constructed via solving a Monge-Ampère type PDE.

Method guarantees regularity of the hypersurface.

Abstract

Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M. We look for hypersurfaces of E with a prescribed vertical Gaussian curvature. In trying to solve this problem fibre-wise, we loose the regularity of the resulting solution. To unsure the smoothness of the solution, we construct it as a radial graph over the unit sphere subbundle of E and prove its existence by solving in this one a nonlinear partial differential equation of Monge-Amp\`ere type.