arXiv research

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …

Paper defines a new functional for spinors on Euclidean manifolds.

Proves positive mass theorem for specific manifold types.

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…

Study the positive mass theorem for certain asymptotic manifolds.

Paper presents a new Pohozaev-Schoen identity for non-compact manifolds.

We present a method in nonlinear elliptic systems to study curvature decays on asymptotically locally Euclidean (ALE) manifolds. In particular, we show that scalar flat Kahler and harmonic ALE metrics of real dimension n are of order n-2.

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.

New ADM mass definition for weakly regular manifolds.

Study on blow-up solutions for semilinear wave equations on specific manifolds.

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.

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…

In this paper, we prove that if an asymptotically Euclidean manifold $(M^n,g)$ under the condition that $R \ge 0$ has long time existence of Ricci flow, the mass of $(M^n,g)$ is nonnegative. In addition, we give an independent proof of positive mass theorem in dimension $3$.

We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length.

The Yamabe flow converges to a specific function on compactified manifolds.

Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…

We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.

The paper proves conditions for isoperimetric regions in curved spaces.

The paper proves rigidity for shells in non-Euclidean spaces.

In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constra…

This paper constructs charged Riemannian manifolds to test Penrose inequality.

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Ise…

Paper proves rigidity theorems for AE Q-singular spaces.

Optimizes Dirac eigenvalue bound using curvature and quasi-spherical metrics.

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Q…

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-Horowitz-Maeda.

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional case where it remains open if one of the theoretically possible cones can actuall…

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on …

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical…

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists …

Log-Sobolev inequality proven for submanifolds in specific types of manifolds.

The study shows that certain nearly G2 and nearly Kähler conifolds cannot be resolved by gluing asymptotically conical G2 and Calabi-Yau manifolds.

Alternative approach to rigidity of high-dimensional isometric immersions.

Study on Einstein manifolds linking stability and rigidity.

We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresp…

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

Solves geodesic equations on special Kähler manifolds, proving global regularity.

Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.

The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.

We show that the causal-future-directed character of the energy-momentum vector of $n$-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, $n\ge 3$, can be traced back to that of asymptotically Euclidean general-relativistic initial data sets satisfying the dominant energy cond…

Generalising a proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of the hyperbolic mass near the hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

We prove that any asymptotically Euclidean metric on $\mathbb{R}^n$ with no conjugate points must be isometric to the Euclidean metric.

In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds with curvature decay and controlled holonomy. As a…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n und…

Develops a new asymptotic efficiency theory for non-Euclidean parameter spaces.

This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic conditi…

We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold…