Lan-Hsuan Huang

On the center of mass of isolated systems with general asymptotics

We propose a definition of center of mass for asymptotically flat manifolds satisfying the Regge–Teitelboim condition at infinity. This definition has a coordinate-free expression and natural properties. Furthermore, we prove that our definition is consistent both with the one proposed by Corvino and Schoen and another by Huisken and Yau. The main tool is a new density theorem for data satisfying the Regge–Teitelboim condition.

The equality case of the Penrose inequality for asymptotically flat graphs

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main theorem states that if the equality holds, then the hypersurface is a Schwarzschild solution. As part of our proof, we show that asymptotically flat graphical hypersurfaces with a minimal boundary and non-negative scalar curvature must be mean convex, using the argument that we developed earlier. This enables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation.

The spacetime positive mass theorem in dimensions less than eight

We prove the spacetime positive mass theorem in dimensions less than eight. This theorem states that for any asymptotically flat initial data set satisfying the dominant energy condition, the ADM energy-momentum vector

Specifying angular momentum and center of mass for vacuum initial data sets

(E,P)

Solutions of special asymptotics to the Einstein constraint equations

of the initial data satisfies the inequality

Stability of the Positive Mass Theorem for Graphical Hypersurfaces of Euclidean Space

E \ge |P|

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

. Previously, this theorem was proven only for spin manifolds by E. Witten. Our proof is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author. An important part of our proof is to introduce an appropriate substitute for the area functional used in the time-symmetric case. We also establish a density theorem of independent interest that allows us to reduce the general case of our theorem to the case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.

Hypersurfaces with nonnegative scalar curvature

We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.

Rigidity theorems on hemispheres in non-positive space forms

We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are ill-defined.

Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in