论文标题
积极曲率表面上的对数洞穴和基本间隙
Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature
论文作者
论文摘要
我们研究了凸域的Laplacian的第一个Dirichlet本征函数的对数concovity。对于满足涉及曲率及其第二个衍生物的条件的正面弯曲表面，我们表明第一个特征功能是强烈的对数 - concove。以前，对于一般凸域而言，只有躺在$ \ m mathbb {r}^n $和$ \ mathbb {s}^n $中时，才知道第一个特征functions的log-concovity。使用此估计，我们在此类地区的基本差距上建立了较低的界限。此外，我们研究了在RICCI流量和度量标准的其他变形下这些估计值的行为。
We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first eigenfunction is strongly log-concave. Previously, for general convex domains, the log-concavity of the first eigenfunctions were only known when lying in $\mathbb{R}^n$ and $\mathbb{S}^n$. Using this estimate, we establish lower bounds on the fundamental gap of such regions. Furthermore, we study the behavior of these estimates under Ricci flow and other deformations of the metric.
