The Nirenberg problem of prescribed Gauss curvature on $S^{2}$

TL;DR

This paper offers a new approach to the Nirenberg problem by employing the Cheeger-Gromov compactness theorem, establishing a proper Fredholm map structure, and deriving new existence and non-existence results without relying on Sobolev inequalities.

Contribution

It introduces a novel perspective using compactness techniques, proves properness of the curvature map, and generalizes the Moser theorem, advancing understanding of prescribed Gauss curvature on the sphere.

Findings

Proper Fredholm map structure for the curvature map in stable regions

New existence and non-existence results for prescribed Gauss curvature

Generalization of Moser's theorem on conformal metrics

Abstract

We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures $K$ contained in naturally defined stable regions. We prove that in such stable regions, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on $S^{2}$. In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger-Moser-Aubin-Onofri.