Luen-Fai Tam

PSEUDOLOCALITY FOR THE RICCI FLOW AND APPLICATIONS

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.

On the volume functional of compact manifolds with boundary with constant scalar curvature

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 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Rigidity of compact manifolds and positivity of quasi-local mass

In this paper, we obtain some rigidity theorems on compact manifolds with nonempty boundary. The results may be related to the positivity of some quasi-local mass of Brown?York type. The main argument is to use monotonicity of quantities similar to the Brown?York quasi-local mass in a foliation of quasi-spherical metrics. Together with a hyperbolic version of positivity of a mass quantity, we obtain our main results.

Einstein and conformally flat critical metrics of the volume functional

Let R be a constant. Let M R γ be the space of smooth metrics g on a given compact manifold Ω n (n > 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 ∈ M R γ . In this work, we classify all Einstein or conformally flat metrics which are critical points of V(·) in M R γ .

On the elliptic equation Δ+-^{}=0 on complete Riemannian manifolds and their geometric applications

We study the elliptic equation ∆u + ku −Kup = 0 on complete noncompact Riemannian manifolds with K nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space Rn and the hyperbolic space Hn are carried out when k is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.

Critical Points of Wang–Yau Quasi-Local Energy

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 ≤ H0 where H is the mean curvature of Σ in Ω and H0 is the mean curvature of Σ when isometrically embedded in

On Second Variation of Wang-Yau Quasi-Local Energy

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A C 0 -estimate for the parabolic Monge–Ampère equation on complete non-compact Kähler manifolds

. If Ω is not isometric to a domain in