A general existence proof for non-linear elliptic equations in semi-Riemannian spaces

TL;DR

This paper provides a broad existence proof for non-linear elliptic equations in semi-Riemannian spaces, applicable to complex geometric problems without requiring curvature sign conditions or uniqueness assumptions.

Contribution

It introduces a general existence framework for non-linear elliptic equations in semi-Riemannian spaces, applicable to curvature-prescribed hypersurface problems without sign restrictions.

Findings

Existence of solutions for curvature-prescribed hypersurfaces in Riemannian manifolds.

Applicable to problems with barrier conditions, independent of uniqueness assumptions.

Method does not rely on curvature flow or successive approximation.

Abstract

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold $N$ without supposing any sign condition on the sectional curvatures $K_N$. A curvature flow wouldn't work in this situation, neither the method of successive approximation.