The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics

TL;DR

This paper investigates the degree of uniform ellipticity in fully nonlinear second order equations by analyzing eigenvalues, leading to solutions for nonlinear geometric problems like the Loewner-Nirenberg and Yamabe problems.

Contribution

It introduces a novel approach to assess partial uniform ellipticity and applies it to solve advanced nonlinear geometric PDEs with topological considerations.

Findings

Established a method to measure proximity to uniform ellipticity

Solved a nonlinear Loewner-Nirenberg problem

Addressed a noncompact nonlinear Yamabe problem

Abstract

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.