arXiv research

Existence proved for static vacuum extensions near Schwarzschild spheres.

Revises Schwarzschild manifold rigidity proof for spin manifolds.

Study on area-constrained Willmore spheres in asymptotic Schwarzschild manifolds.

The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.

Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.

The paper proves the stability of a flow in Schwarzschild space.

Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.

Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…

In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass.

Adapting Israel's proof of static black hole uniqueness, we show that the Schwarzschild spacetime is the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere.

Hopf's theorem generalized to curved spaces.

The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.

We solve Bartnik's stationary extension problem near Schwarzschild spheres.

Let $(M, g)$ be a complete Riemannian $3$-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We \textsl{unconditionally} characterize the large, embedded stable constant mean curvature spheres in $(M, g)$.

The paper establishes inequalities for $p$-capacitary functions in flat half-spaces.

We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically …

We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on th…

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}=(1+\frac{1}{l})^{2}δ_{ij}+O(l^{-2})$. The existence of uns…

We study rigidity of minimal two-spheres $Σ$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $Σ$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\…

Study on static manifolds with boundary and rigidity of curvature.

Paper proves rigidity of static manifolds and applies to metric extensions.

Constructs surfaces with constant mean curvature in Schwarzschild spacetime near null infinity.

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all suffi…

We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is $C^{3}-$close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such an end is foliated by spheres of Willmore type. In this paper, we prove that the leaves of this foliation are stable under sm…

The paper derives inequalities for $p$-capacitary functions in 3-manifolds with nonnegative scalar curvature.

Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken-Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard [t=0]-timeslice of the Schwarzschild solution. In this paper, w…

We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological consta…

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $Σ$ is strictly mean convex and star-shaped, then the flow hypersurface $Σ_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an a…

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

We show a uniqueness result for the n-dimensional spatial Reissner-Nordström manifold: a static, electrovacuum, asymptotically flat system which is asymptotically Reissner-Nordström is a subextremal Reissner-Nordström manifold with positive mass, provided that its inner boundary is a (possibly disconnected) photon sphe…

The study finds regular null hypersurfaces in a perturbed Schwarzschild black hole exterior.

Study gluing event horizons of Minkowski and Schwarzschild spacetimes.

Study finds new minimal surfaces in Schwarzschild space.

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that th…

Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.

The paper studies static manifolds with boundary and their properties.

Two rigidity results for surfaces in Schwarzschild spacetime.

We prove a version the Penrose inequality for black hole space-times which are perturbations of the Schwarzschild exterior in a slab around a null hypersurface $\underline{\mathcal{N}}_0$. $\underline{\mathcal{N}}_0$ terminates at past null infinity $\mathcal{I}^-$ and $\mathcal{S}_0:=\partial\underline{\mathcal{N}}_0$…

Higher-dimensional Schwarzschild spacetimes violate the Penrose property.

Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.

In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term …

Schwarzschild 3-manifold stability proven for 3D Penrose inequality.

Complete minimal surfaces with any genus found in a specific 3-manifold.

Proves stability of Schwarzschild black holes without symmetry assumptions.

Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.

Study on spacelike submanifolds in generalized Schwarzschild spacetimes with lightlike foliations.