arXiv research

Revises Schwarzschild manifold rigidity proof for spin manifolds.

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

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

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

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

Paper defines semi-quasi-Einstein manifolds and applies to Schwarzschild and Kottler spacetimes.

Existence proved for static vacuum extensions near Schwarzschild spheres.

In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.

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

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

We study the problem of existence of isoperimetric regions for large volumes, in $C^0$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in …

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

Proves equality in Minkowski inequality for static, flat manifolds.

In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.

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 note we address the problem of finding Abelian instantons of finite energy on the Euclidean Schwarzschild manifold. This amounts to construct self-dual L^2 harmonic 2-forms on the space. Gibbons found a non-topological L^2 harmonic form in the Taub-NUT metric, leading to Abelian instantons with continuous energ…

We will extend partially our previous results about the limit of the Brown-York mass of a family of convex revolution surfaces in the Schwarzschild manifold such that these surfaces may have unbounded ratios of their radii.

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

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

In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.

The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.

We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.

Study on static manifolds with boundary and rigidity of curvature.

Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.

We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold $(M^n,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactifi…

Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.

The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R^n; for example, such an ine…

Hopf's theorem generalized to curved spaces.

Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.

We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose ineq…

Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat 3-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manif…

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

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, general…

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

Study finds new minimal surfaces in Schwarzschild space.

In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Using their method, Rigger proved the same theorem for Riemannian manifold…

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extens…

The first irreducible solution of the $\mathrm{SU} (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\mathbb{R}^4$ were discovered. While soon after, in 1978, the ADHM construction gave a complete descrip…

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

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…

Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only…

Two rigidity results for surfaces in Schwarzschild spacetime.

Higher-dimensional Schwarzschild spacetimes violate the Penrose property.

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 …

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

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space $\R^n$, the hyperbolic space $\H^n$ and a Riemannian manifold $\mathfrak{S^n}$ ($n\geq 3$) with the Schwarzschild metric to any Riemannian manifold $N$.

Proves stability of Schwarzschild black holes without symmetry assumptions.

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)\…