Toroidal marginally outer trapped surfaces in closed Friedmann-Lemaitre-Robertson-Walker spacetimes: Stability and isoperimetric inequalities

We investigate toroidal Marginally Outer Trapped Surfaces (MOTS) and Marginally Outer Trapped Tubes (MOTT) in closed Friedmann-Lemaitre-Robertson-Walker (FLRW) geometries. They are constructed by embedding Constant Mean Curvature (CMC) Clifford tori in a FLRW spacetime. This construction is used to assess the quality of certain isoperimetric inequalities, recently proved in axial symmetry. Similarly to spherically symmetric MOTS existing in FLRW spacetimes, the toroidal ones are also unstable.Comment: 7 pages, 2 figure

Construction of vacuum initial data by the conformally covariant split system

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split system defined on compact manifolds with or without boundaries. In the former case, the boundary corresponds to an apparent horizon in the constructed initial data. The case with a cosmological constant is then considered separately. Finally, to demonstrate the applicability of the conformal covariant split system in numerical studies, we provide numerical examples of solutions on manifolds $\mathbb S^1 \times \mathbb S^2$ and $\mathbb S^1 \times \mathbb T^2$

On Hawking mass and Bartnik mass of CMC surfaces

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces.Comment: version accepted by Math. Res. Let

Quasi-local mass integrals and the total mass

On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.Comment: References updated, introduction revise

Bartnik mass via vacuum extensions

We construct asymptotically flat, scalar flat extensions of Bartnik data $(\Sigma, \gamma, H)$, where $\gamma$ is a metric of positive Gauss curvature on a two-sphere $\Sigma$, and $H$ is a function that is either positive or identically zero on $\Sigma$, such that the mass of the extension can be made arbitrarily close to the half area radius of $(\Sigma, \gamma)$. In the case of $H \equiv 0$, the result gives an analogue of a theorem of Mantoulidis and Schoen, 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 $(\Sigma, \gamma)$, 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 \cite{ShiTam02} and a refined Shi-Tam monotonicity, found by the first named author in \cite{Miao09}.Comment: references updated; to appear in International Journal of Mathematics, a special issue dedicated to the mathematical contribution of Professor Luen-Fai Ta

Positive mass theorems for asymptotically AdS spacetimes with arbitrary cosmological constant

We formulate and prove the Lorentzian version of the positive mass theorems with arbitrary negative cosmological constant for asymptotically AdS spacetimes. This work is the continuation of the second author's recent work on the positive mass theorem on asymptotically hyperbolic 3-manifolds.Comment: 17 pages, final version, to appear in International Journal of Mathematic