Diameter theorems on K\"ahler and quaternionic K\"ahler manifolds under a positive lower curvature bound

TL;DR

This paper introduces the orthogonal Bakry-Émery tensor as a generalization of orthogonal Ricci curvature and establishes sharper diameter bounds for Kähler and quaternionic Kähler manifolds under positive lower curvature bounds.

Contribution

It defines the orthogonal Bakry-Émery tensor and proves new diameter theorems with improved bounds for Kähler and quaternionic Kähler manifolds under positivity assumptions.

Findings

Sharper Bonnet-Myers diameter bounds under orthogonal Bakry-Émery curvature positivity.

Extension of diameter theorems to Kähler and quaternionic Kähler geometries.

Generalization of curvature conditions leading to diameter estimates.

Abstract

We define the orthogonal Bakry-\'Emery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on K\"ahler and quaternionic K\"ahler manifolds under positivity assumption on the orthogonal Bakry-\'Emery tensor. Moreover, under such assumptions on the orthogonal Bakry-\'Emery tensor and the holomorphic or quaternionic sectional curvature on a K\"ahler manifold or a quaternionic K\"ahler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.