arXiv research

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

Minimal surfaces in Schwarzschild manifold have Morse index one.

Revises Schwarzschild manifold rigidity proof for spin manifolds.

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

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 …

Proves non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics.

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

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

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…

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…

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

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 $N$.

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

Study on static manifolds with boundary and rigidity of curvature.

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 $(M, g)$.

New inequalities linking manifold capacities and quasi-local masses derived.

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…

Hopf's theorem generalized to curved spaces.

Study proves non-degeneracy of certain metrics in linearized gravity.

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric e…

Proves rigidity for surfaces in Schwarzschild manifold.

Proves Penrose inequality in all dimensions for specific manifolds.

We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.

Study mean curvature flow solitons in warped products and Riemannian manifolds.

Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.

Smooth metrics satisfying Penrose inequality are necessarily smooth.

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)\…

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

Study on stability of mass theorems using foliated IMCF.

Paper extends foliation results in higher dimensions for Schwarzschild spaces.

The paper studies static manifolds with boundary and their properties.

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

Existence proved for static vacuum extensions near Schwarzschild spheres.

Einstein manifolds are rigid under certain metric deformations.

We show that the Riemannian Schwarzschild and the ``Taub-bolt'' instanton solutions are the only spaces (M,g) such that 1) M is a 4-dimensional, simply connected manifold with a Riemannian, Ricci-flat C^2-metric g which admits (at least) a 1-parameter group of isometries H without isolated fixed points on M. 2) The quo…

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…

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

Study of area preserving Willmore flow on Schwarzschild-like manifolds.

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 rigidity of minimal disks in specific 3-manifolds.

Proves equality in Minkowski inequality for static, flat 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 a paper \cite{P} in 1973, R. Penrose made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16π}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemann…

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.

The paper proves conditions for stable constant mean curvature surfaces in specific manifolds.

We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we …

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…

The paper studies harmonic forms and spinors on Taub-bolt space.