4 papers
Gradient bounds for viscosity solutions to certain elliptic equations
Thalia Jeffres, Xiaolong Li
Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function \( u: \mathbb{R} ^{n} \rightarrow \mathbb{R} \) which satisfies a degenerat…
Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem
Xiaolong Li
We prove that if a closed Riemannian manifold has finite fundamental group and satisfies the curvature condition \begin{equation*} R_{1313} +R_{1414} +R_{2323} + R_{2424}…
New Sphere Theorems under Curvature Operator of the Second Kind
Xiaolong Li
We investigate Riemannian manifolds whose curvature operator of the second kind satisfies the condition \begin{equation*} α^{-1} (λ_1 +\cdots +λ_α) > - θ\b…
The curvature operator of the second kind in dimension three
Harry Fluck, Xiaolong Li
This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvatur…