Geometric inequalities on Closed Surfaces with $λ_1(-Δ+β K)\geq0$

Kai Xu

Published 2022-11-21Version 1

Let $\Sigma$ be a closed orientable surface satisfying the eigenvalue condition $\lambda_1(-\Delta+\beta K)\geq\lambda\geq0$, where $\beta$ is a positive constant and $K$ is the Gaussian curvature of $\Sigma$. This eigenvalue condition naturally arises for stable minimal surfaces and $\mu$-bubbles in 3-manifolds with positive scalar curvature (with $\beta=1$ there). We prove that when $\beta>1/2$, the Cheeger constant of $\Sigma$ is bounded from below by $C(\beta)/diam(\Sigma)$. When $\beta>1/4$, the homogeneous isoperimetric ratio of $\Sigma$ is bounded from below by $C(\beta,\epsilon)\big(\frac{Area(\Sigma)}{diam(\Sigma)^2}\big)^{\frac{1+\epsilon}{4\beta-1}}$ ($\forall\epsilon>0$). We also show a weak Bonnet-Myers' theorem $diam(\Sigma)\leq C(\beta)\lambda^{-1/2}$ and a total volume comparison theorem $Area(\Sigma)\leq C(\beta)\,diam(\Sigma)^2$, when $\beta>1/4$. Generalizing Schoen-Yau's observation, associated to each positive supersolution $\Delta\varphi\leq\beta K\varphi$ we make a conformal change $\tilde g=\varphi^{2/\beta}g$ with $K_{\tilde g}\geq0$. Some of the main theorems are proved by studying the geometric inequalities for $\tilde g$ and transferring them back to $g$.