Metrics On $S^2$ With Bounded $\|K_g\|_{L^1\log L^1}$ And Small $\|K_g-1\|_{L^1}$

TL;DR

This paper investigates the convergence properties of metric sequences on the 2-sphere with specific bounds on their Gaussian curvature, demonstrating precompactness under certain integral conditions.

Contribution

It establishes precompactness of metric sequences on S^2 with bounded |K_g|\,log(1+|K_g|) and small |K_g-1|, a novel result in geometric analysis.

Findings

Sequence is precompact under given bounds.

Bounded |K_g|\,log(1+|K_g|) controls curvature behavior.

Small |K_g-1| indicates convergence to the standard sphere.

Abstract

In this short paper, we will study the convergence of a metric sequence $g_k$ on $S^2$ with bounded $\int|K_{g_k}|\log(1+|K_{g_k}|)d\mu_{g_k}$ and small $\int|K(g_k)-1\|d\mu_{g_k}$. We will show that such a sequence is precompact.