arXiv subjects

Hai-Ping Fu

Publications and source records attributed to Hai-Ping Fu.

6 recordsLinked to original sources

arXiv2026

K{\"a}hler Einstein manifolds and the Calabi curvature operator

In this paper, we study the Calabi curvature operator on K{\"a}hler manifolds. First, we prove that if the Calabi curvature operator on K{\"a}hler manifolds satisfies $\frac{{n\left(n + 1\right)}}{2}$-positive (nonnegative), $\frac{{n + 1}}{2}$-positive (nonnegative), and $\left( n-1 \right)$-positive (nonnegative), then the scalar curvature, Ricci curvature, and orthogonal Ricci curvature are positive (nonnegative), respectively. Second, we show that any compact K{\"a}hler Einstein manifold satisfying the condition $$\lambda_1+\dots+\lambda_{\alpha}\ge -{\alpha}\theta(n,{\alpha})\bar\lambda,\; {\alpha}\le \frac{n}{2}$$ must have nonnegative constant holomorphic sectional curvature.

math.DG↗

arXiv2023

Einstein manifolds and curvature operator of the second kind

We prove that a compact Einstein manifold of dimension $n\geq 4$ with nonnegative curvature operator of the second kind is a constant curvature space by Bochner technique. Moreover, we obtain that compact Einstein manifolds of dimension $n\geq 11$ with $\left [ \frac{n+2}{4} \right ]$-nonnegative curvature operator of the second kind, $4\ (\mbox{resp.},8,9,10)$-dimensional compact Einstein manifolds with $2$-nonnegative curvature of the second kind and $5$-dimensional compact Einstein manifolds with $3$-nonnegative curvature of the second kind are constant curvature spaces. Combing with Li's result [10], we have that a compact Einstein manifold of dimension $n\geq 4$ with $\max\{4,\left [ \frac{n+2}{4} \right ]\}$-nonnegative curvature operator of the second kind is a constant curvature space.

math.DG↗

arXiv2016

Four manifolds with postive Yamabe constant

We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality. We prove some classification theorems of four manifolds according to some conformal invariants (see Theorems 1.3 and 1.6), which generalize the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}.

math.DG↗

arXiv2016

Some $L^p$ rigidity results for complete manifolds with harmonic curvature

Let $(M^n, g)(n\geq3)$ be an $n$-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by $R$ and $\mathring{Rm}$ the scalar curvature and the trace-free Riemannian curvature tensor of $M$, respectively. The main result of this paper states that $\mathring{Rm}$ goes to zero uniformly at infinity if for $p\geq \frac n2$, the $L^{p}$-norm of $\mathring{Rm}$ is finite. Moreover, If $R$ is positive, then $(M^n, g)$ is compact. As applications, we prove that $(M^n, g)$ is isometric to a spherical space form if for $p\geq \frac n2$, $R$ is positive and the $L^{p}$-norm of $\mathring{Rm}$ is pinched in $[0,C_1)$, where $C_1$ is an explicit positive constant depending only on $n, p$, $R$ and the Yamabe constant. In particular, we prove an $L^{p}(\frac n2\leq p<\frac{n-2}{2}(1+\sqrt{1-\frac4n}))$-norm of $\mathring{Ric}$ pinching theorem for complete, simply connected, locally conformally flat Riemannian $n(n\geq 6)$-manifolds with constant negative scalar curvature. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian $n$-manifolds with constant positive scalar curvature, which improves Thereom 1.1 and Corollary 1 of E. Hebey and M. Vaugon \cite{HV}. This rsult is sharped, and we can precisely characterize the case of equality.

math.DG↗

arXiv2016

On compact manifolds with harmonic curvature and positive scalar curvature

Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4, Corollary 1.6 and Theorem 1.9 are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality.

math.DG↗

arXiv2015

Rigidity Theorem for integral pinched shrinking Ricci solitons

We prove that an $n$-dimensional, $n\geq4$, compact gradient shrinking Ricci soliton satisfying a $L^{\frac n2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^n$, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).

math.DG↗