Work

New sphere theorems under curvature operator of the second kind

Year: 2025

Type: article

Abstract: We investigate Riemannian manifolds whose curvature operator of the second kind satisfies the condition where are the eigenvalues of , is their average, and . Under such curvature conditions with optimal (depending on and ), we prove differentiable sphere theorems and their rigidity results in dimensions 3 and 4, a homological sphere theorem in higher dimensions, and a curvature characterization of Kähler space forms. These results generalize recent works corresponding to (namely, ‐positivity of ) of Cao–Gursky–Tran, Nienhaus–Petersen–Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of most results.

Source: Journal of the London Mathematical Society

DOI: 10.1112/jlms.70356

Author: Xiaolong Li

Institution: Auburn University

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funder: National Science Foundation

Awards: LEAPS-MPS: Geometrization with Positive Curvature, Sectional Curvature, Isotropic Curvature, and Curvature Operators