Conformal metric sequences with integral-bounded scalar curvature

TL;DR

This paper investigates the convergence behavior of conformal metrics with bounded volume and scalar curvature in an integral sense on compact manifolds, using advanced inequalities and the 3-circle theorem.

Contribution

It introduces a novel approach combining the 3-circle theorem and John-Nirenberg inequality to analyze bubble tree convergence of conformal metrics.

Findings

Established conditions for bubble tree convergence under integral bounds.

Applied the 3-circle theorem and John-Nirenberg inequality in geometric analysis.

Provided new insights into scalar curvature behavior in conformal metric sequences.

Abstract

Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will use the 3-circle theorem and the John-Nirenberg inequality to study the bubble tree convergence of $g_{k}$.