arXiv subjects

Haiqing Cheng

Publications and source records attributed to Haiqing Cheng.

5 recordsLinked to original sources

arXiv2026

Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds

In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ \alpha^{-1}\big(\sum_{i=1}^{[\alpha]} \lambda_i+ (\alpha - [\alpha] ) \lambda_{[\alpha] + 1} \big) \ge -\theta \bar{\lambda} \] for some real number $\alpha \in [1, (n+2)(n-1)/2)$. Here $[\alpha] :=\max\{ m \in \mathbb{Z}: m \leq \alpha\}$, $\theta>-1$ and $\lambda_1 \le \cdots \le \lambda_{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\bar{\lambda}$ is their average. The main result states that any closed Einstein manifold of dimension $n \ge 4$ with $\mathring{R}$ satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to $\alpha \in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.

math.DG↗

arXiv2025

Einstein manifolds under cone conditions for the curvature operator of the second kind

It is established in [6, 14, 23] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [18]. Precisely, we prove that any closed Einstein manifold of dimension $n=4$ or $n=5$ or $n\ge 8$, if the curvature operator of the second kind $\mathring{R}$ satisfies \begin{align*} (\lambda_1+\lambda_2)/2 \ge -\theta(n) \bar \lambda, \end{align*} then the manifold is either flat or a round sphere. Here, $\lambda_1\le \lambda_2\le \cdots\le \lambda_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $ \bar \lambda $ is their average, and $\theta(n)$ is a positive constant defined as in (1.2).

math.DG↗

arXiv2024

Einstein manifolds of negative lower bounds on curvature operator of the second Kind

We demonstrate that $n$-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ satisfies $\lambda_1 \ge -\theta(n) \bar\lambda$, are either flat or round spheres, where $\bar \lambda$ is the average of the eigenvalues of $\mathring{R}$, and $\theta(n)$ is defined as in equation (1.2). Our result improves a celebrated result (Theorem 1.1) concerning Einstein manifolds with nonnegative curvature operator of the second kind.

math.DG↗

arXiv2022

Comparison results for Poisson equation with mixed boundary condition on manifolds

In this article, we establish a $L^1$ estimate for solutions to Poisson equation with mixed boundary condition, on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. On Riemann surfaces we obtain a Talenti-type comparison. Our results generalize main theorems in [2] to Riemannian setting, and Chen-Li's result [8] to the case of variable Robin parameter.

math.DG↗

arXiv2021

Schwarz symmetrizations in parabolic equations on complete manifolds

In this article, we prove a sharp estimate for the solutions to parabolic equations on manifolds. Precisely, using symmetrization techniques and isoperimetric inequalities on Riemannian manifold, we obtain a Bandle's comparison on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. Our results generalize Bandle's result [6] to Riemannian setting, and Talenti's comparison for elliptic equation on manifolds by Colladay-Langford-McDonald [12] and Chen-Li [9] to parabolic equations.

math.DG↗