Related papers: Noncompact Fill-Ins of Bartnik Data

We show that a closed non-orientable $3$-manifold admits a positive scalar curvature metric if and only if its orientation double cover does; however, for each $4\le n\le 7$, there exist infinitely many smooth non-orientable $n$-manifolds…

We study complete, finite volume $n$-manifolds $M$ of bounded nonpositive sectional curvature. A classical theorem of Gromov says that if such $M$ has negative curvature then it is homeomorphic to the interior of a compact…

In this paper, we are able to prove an analogy of the Calabi-Yau theorem for complete Riemannian manifolds with nonnegative scalar curvature which are aspherical at infinity. The key tool is an existence result for arbitrarily large bounded…

In this paper we confirm a folklore conjecture which suggests that for a complete noncompact manifold $M$ of finite volume with sectional curvature $-1 \leq K \leq 0$, if the universal cover of $M$ is a visibility manifold, then the…

Let $(M,g^{TM})$ be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and $F$ an integrable subbundle of $T M$ . Let $k^F$ be the leafwise scalar curvature associated to $g^F=g^{TM}|_F$.…

Given a metric $\gamma$ of nonnegative Gauss curvature and a positive function $H$ on a $2$-sphere $\Sigma$, we estimate the Bartnik quasi-local mass of $(\Sigma, \gamma, H)$ in terms of the area, the total mean curvature, and a quantity…

In this work, we show that complete non-compact manifolds with non-negative Ricci curvature, Euclidean volume growth and sufficiently small curvature concentration are necessarily flat Euclidean space.

In this article we extend several foundational results of the theory of complete minimal surfaces of finite index in the Euclidean space to minimal surfaces in asymptotically flat manifolds and, more generally, to marginally outer-trapped…

The Positive Mass Conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the Positive Mass Conjecture states that in the above situation, if the…

In this work, we study a gap phenomenon in locally conformally flat Riemannian manifolds with non-negative Ricci curvature. We construct complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve.…

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…

We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and $L^2$ harmonic forms. In the case where the manifold is the universal cover of a…

We prove two rigidity theorems for open (complete and noncompact) $n$-manifolds $M$ with nonnegative Ricci curvature and the infimum of volume growth order $<2$. The first theorem asserts that the Riemannian universal cover of $M$ has…

Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally…

We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities. Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the…

We consider a non-negative biminimal properly immersed submanifold $M$ (that is, a biminimal properly immersed submanifold with $\lambda\geq0$) in a complete Riemannian manifold $N$ with non-positive sectional curvature. Assume that the…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using…

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit…

The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the…