Comparison theorems in Hermitian geometry I

TL;DR

This paper extends classical comparison theorems from Riemannian geometry to Hermitian geometry by developing second variational formulas and index forms, leading to new geometric inequalities and results.

Contribution

It introduces second variational formulas and index forms in Hermitian geometry, establishing analogues of Myers' theorem, Laplacian, and volume comparison theorems.

Findings

Hermitian geometry comparison theorems established

Myers' theorem analogue proved

Laplacian and volume comparison results derived

Abstract

This paper develops second variational formulas and index forms in the context of Hermitian geometry. Building upon these analytical foundations, we establish results analogous to classical theorems in Riemannian geometry, including Myers' theorem, Laplacian comparison theorems and volume comparison theorems.