arXiv subjects
Publications and source records attributed to Luen-Fai Tam.
Motivated by the results of Wu-Yau-Zheng \cite{WuYauZheng}, we show that under a certain curvature assumption the harmonic representative of any boundary class of the Kähler cone is nonnegative.
In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $Σ$ has positive Gaussian curvature and all boundary components of $Ω$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $Σ$ in $Ω$ and $H_0$ is the mean curvature of $Σ$ when isometrically embedded in $R^3$. If $Ω$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $Σ$ in $Ω$ is a strict local minimum of the Wang-Yau quasi-local energy of $Σ$, 2. on a small perturbation $\tildeΣ$ of $Σ$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tildeΣ$.
In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an asymptotically flat manifold.
In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang .
We study a parabolic complex Monge-Ampère type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form $Ω$ and initial \K metric $g_0$ on $M$, the modified \KR flow $g'=-\Ric+Ω$ has a long time smooth solution converging to a complete \K metric such that $\Ric=Ω$, which extends the result in [1] to non-compact manifolds. We will also obtain a long time existence result for the \KR flow which generalizes a result [5].
Let $M=X\times Y$ be the product of two complex manifolds of positive dimensions. In this paper, we prove that there is no complete Kähler metric $g$ on $M$ such that: either (i) the holomorphic bisectional curvature of $g$ is bounded by a negative constant and the Ricci curvature is bounded below by $-C(1+r^2)$ where $r$ is the distance from a fixed point; or (ii) $g$ has nonpositive sectional curvature and the holomorphic bisectional curvature is bounded above by $-B(1+r^2)^{-δ}$ and the Ricci curvature is bounded below by $-A(1+r^2)^γ$ where $A, B, γ, δ$ are positive constants with $γ+2δ<1$. These are generalizations of some previous results, in particular the result of Seshadri and Zheng.
Given a spacelike 2-surface $Σ$ in a spacetime $N$ and a constant future timelike unit vector $T_0 $ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(Σ, X, T_0)$ for a given isometric embedding $X$ of $Σ$ into a flat 3-slice in $\R^{3,1}$. The quantity $ E(Σ, X, T_0) $ itself depends on the choice of $X$, however the infimum of $ E(Σ, X, T_0)$ over $ T_0 $ does not. In particular, when $Σ$ lies in a time symmetric 3-slice in $N$ and has nonnegative Brown-York quasilocal mass $\mby(Σ)$, our estimates show that $\inf\limits_{T_0}E(Σ, X, T_0)$ equals $ \mby (Σ)$. We also study the spatial limit of $ \inf\limits_{T_0}E(S_r,X_r,T_0)$, where $S_r$ is a large coordinate sphere in a fixed end of an asymptotically flat initial data set $(M, g, p)$ and $X_r$ is an isometric embeddings of $S_r$ into $\mathbb{R}^3 \subset \mathbb{R}^{3,1}$. We show that if $(M, g, p)$ has future timelike ADM energy-momentum, then $\lim\limits_{r\to\infty}\inf\limits_{T_0}E(S_r,X_r,T_0)$ equals the ADM mass of $(M, g, p)$.
We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau \cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed 2 dimensional surfaces evolving in an ambient three dimensional manifold. As an interesting by-product, we are able to write the ADM mass \cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere $S_r$ and an integral of the scalar curvature plus a geometrically constructed function $Φ(x)$ in the asymptotic region outside $S_r $. In the third part, we prove that for any closed, spacelike, 2-surface $Σ$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is defined, if $Σ$ bounds a compact spacelike hypersurface in $\R^{3,1}$, then the Liu-Yau mass of $Σ$ is strictly positive unless $Σ$ lies on a hyperplane. We also show that the examples given by Ó Murchadha, Szabados and Tod \cite{MST} are special cases of this result.
Let $R$ be a constant. Let $\mathcal{M}^R_γ$ be the space of smooth metrics $g$ on a given compact manifold $Ω^n$ ($n\ge 3$) with smooth boundary $Σ$ such that $g$ has constant scalar curvature $R$ and $g|_Σ$ is a fixed metric $γ$ on $Σ$. Let $V(g)$ be the volume of $g\in\mathcal{M}^R_γ$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $V(\cdot)$ in $\mathcal{M}^R_γ$.
In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete non-compact Kähler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler Ricci flow \eqref{KRF} has a long time smooth solution $g(t)$ converging smoothly uniformly on compact sets to a complete Kähler Ricci flat metric on $M$. The main step is to obtain a uniform $C^0$-estimates for the corresponding parabolic Monge Ampère equation. Our results on this can be viewed as a parabolic version of the main results in \cite{TY3} on the elliptic Monge Ampère equation.
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and ''small'' hyperbolic and spherical balls in dimensions 3 to 5, the standard space form metrics are indeed saddle points for the volume functional.
We study complete noncompact long time solutions $(M, g(t))$ to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\ijb \ge cRg_\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient Kähler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for $(M, g(t))$. Combining our results with Cao's splitting for Kähler Ricci flow \cite{Cao04} and techniques of Ni-Tam \cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat a special cases, complete Kähler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient Kähler Ricci solitons.
In this paper, we will study the limiting behavior of the Brown-York mass of the coordinate spheres in an asymptotically flat manifold. Limiting behaviors of volumes of regions related to coordinate spheres are also obtained, including a discussion on the isoperimetric mass introduced by Huisken \cite{Huisken}. We will also study expansions of the Brown-York mass and the Hawking mass of geodesic spheres with center at a fixed point $p$ of a three manifold. Some geometric consequences will be derived.
Yau's uniformization conjecture states: a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to $\ce^n$. The Kähler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kähler-Ricci flow with focus on its application to uniformization. Other interesting methods and results related to the study of Yau's conjecture are also discussed.
Let $(M^n, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that $M$ is holomorphically covered by a pseudoconvex domain in $\C^n$ which is homeomorphic to $\R^{2n}$, provided $(M^n, g)$ has uniform linear average quadratic curvature decay.
In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the \KRF flow on complete non-negatively curved \K manifolds.
In this paper, by modifying the arguments in \cite{WY}, we get some rigidity theorems on compact manifolds with nonempty boundary. The results in this paper are similar with those in \cite{ST} and \cite{WY}. Like \cite{ST} and \cite{WY}, we still use quasi-spherical metrics introduced by \cite{Ba} to get monotonicity of some quantities.
In this paper, we construct a family of asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to -6. The manifolds we constructed can be arbitrary close to anti-de Sitter-Schwarzschild manifolds at infinity. Hence, the mass of our manifolds can be very large or very small. The main arguments we used in this paper is gluing methods which was used in \cite{M}.