Pengzi Miao

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.

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 capacity of surfaces in manifolds with nonnegative scalar curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ3, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

A remark on boundary effects in static vacuum initial data sets

Let (M, g) be an asymptotically flat static vacuum initial data set with non-empty compact boundary ?. We prove that (M, g) is isometric to a spacelike slice of a Schwarzschild spacetime under the mere assumption that ? has zero mean curvature, hence generalizing a classic result of Bunting and Masood-ul-Alam. In the case that ? has constant positive mean curvature and satisfies a stability condition, we derive an upper bound of the ADM mass of (M, g) in terms of the mean curvature and area of ?. Our discussion is motivated by Bartniks quasi-local mass definition.

On Existence of Static Metric Extensions in General Relativity

Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik [4]. We show that, for any metric on ¯B1 that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists an asymptotically flat and scalar flat static metric extension in M=ℝ3∖B1 such that it satisfies Bartniks geometric boundary condition [4] on ∂B1.

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

A new monotone quantity along the inverse mean curvature flow in ℝn

{\mathbb R^3}

On Second Variation of Wang-Yau Quasi-Local Energy

. If Ω is not isometric to a domain in