Quantitative comparison theorems in Riemannian and K\"ahler geometry

TL;DR

This paper develops sharp quantitative estimates for Laplacian, area, and volume in Riemannian and Kähler manifolds without curvature restrictions, leading to new comparison theorems and classical results under weak assumptions.

Contribution

It introduces new sharp Laplacian estimates and comparison theorems in Riemannian and Kähler geometry under minimal curvature assumptions, extending classical results.

Findings

Sharp Laplacian upper and lower bounds without curvature assumptions

Quantitative comparison theorems for tubes in manifolds

Generalized Bonnet-Myers and Cheng's eigenvalue estimates

Abstract

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and K\"ahler manifolds under weak integral curvature assumptions. We also give some applications, such as a general Bonnet-Myers theorem and Cheng's eigenvalue estimate under weak integral curvature assumptions.