International Journal of Mathematics

Let $B_n$ be the open unit ball in ${ℂ}^n$ and let $(M^m,g)$ be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map $u\in C^2(\overline{B_n},M)$ satisfying the boundary condition that ${\sum }_{i,j=1}^n{z}_i{\bar{z}}_j{u}_{i\bar{j}}=0$ on $\partial B_n$. We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on $B_n$.

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds $M_n$ obtained by blowing up the diagonal of the product of two copies of a $C{P}^n$.

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

We obtain some basic estimates for a Monge–Ampère type equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmüller space of a compact oriented surface of genus $g(\mathrm{\Sigma })>1$ is diffeomorphic to the disk of dimension $6g(\mathrm{\Sigma })-6$. We also give another proof of properness of a certain energy function on the Teichmüller space.

In this paper, we consider the high-order geometric flows of a compact submanifolds $M$ in a complete Riemannian manifold $N$ with $dim(N)=dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on $N$. Precisely, we show that if $m\in \mathbb{N}$ is strictly larger than the integer part of $n/2$ and $\phi (t)$ is an immersion for all $t\in [0,T)$ and if ${𝔉}_m({\phi }_0)$ is bounded by a constant which relies on the injectivity radius $\bar{R}>0$ and sectional curvature ${\bar{K}}_{\pi }({\bar{K}}_{\pi }\le 1)$ of $N$, then $T$ must be $\infty$.

We construct asymptotically flat, scalar flat extensions of Bartnik data $(\mathrm{\Sigma },\gamma ,H)$, where $\gamma$ is a metric of positive Gauss curvature on a two-sphere $\mathrm{\Sigma }$, and $H$ is a function that is either positive or identically zero on $\mathrm{\Sigma }$, such that the mass of the extension can be made arbitrarily close to the half area radius of $(\mathrm{\Sigma },\gamma )$.

In the case of $H\equiv 0$, the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon $(\mathrm{\Sigma },\gamma )$, for any metric $\gamma$ with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality.

The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

We prove estimates interpolating the Schwarz Lemmata of Royden–Yau and the ones recently established by the author. These more flexible estimates provide additional information on (algebraic) geometric aspects of compact Kähler manifolds with nonnegative holomorphic sectional curvature, nonnegative ${Ric}_{\ell }$ or positive $S_{\ell }$.

In this paper, we use spinorial method to define a new quasi-local mass with reference to a Schwarzchild manifold. In the proof of the main theorem, spinors are defined on manifold with two boundary components. Through the Reilly formula, we get the new quasi-local mass is nonnegative. Combined with Riemannian–Penrose inequality, we derive a localized Penrose inequality.

We prove that every locally pluripolar set on a compact complex manifold is pluripolar. This extends similar results in the Kähler case.