Related papers: Static potentials and area minimizing hypersurface…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the…

In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function,…

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether…

In this paper, we focus on Hamilton's pinching conjecture formulated in Hamilton's paper "Three-manifolds with positive Ricci curvature". Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold satisfying the…

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible…

We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a…

In this work, we prove a compactness theorem on the space of all Hamiltonian stationay Lagrangian submanifolds in a compact symplectic manifold with uniform bounds on area and total extrinsic curvature.

We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The…

On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $\epsilon$ tends to zero of solutions $u_{\epsilon}: M \to \mathbb{C}$ to the equation $\Delta u_{\epsilon} + \epsilon^{-2}(1 -…

$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM).…

In this paper we take an approach similar to that in [M] to establish a positive mass theorem for asymptotically hyperbolic spin manifolds admitting corners along a hypersurface. The main analysis uses an integral representation of a…

In [Comm. Anal. Geom., 13(5):845-885, 2005.], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity $(g,\pi)\in H^2\times H^1$. In particular, it was established that the…

There exists in General Relativity an unambiguous notion of Mass associated to asymptotically flat spacetimes known as the ADM mass. The standard expression for the same is a surface integral over spatial infinity of a linear combination of…

We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an…

We prove that the boundary of the trapped region in an asymptotically Euclidean Riemannian manifold of dimension at least 3 is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.

On a class of dynamical spacetimes which are asymptotic as $t\to\infty$ to a stationary spacetime containing a horizon $\mathcal{H}_0$, we show the existence of a unique null hypersurface $\mathcal{H}$ which is asymptotic to…

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge…

Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant…

For asymptotically flat spacetimes, using the inverse mean curvature flow, we show that any compact $2$-surface, $S_0$, whose mean curvature and its derivative for outward direction are positive in spacelike hypersurface with non-negative…

We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support…