Related papers: On $\mathsf{CD}$ spaces with nonnegative curvature…

For a general radially symmetric, non-increasing, non-negative kernel $h\in L ^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with constant nonlocal $h$-mean curvature. Under a suitable "improved integrability"…

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact…

In this paper we use stable capillary surfaces (analogous to the $\mu$-bubble construction) to study manifolds with strictly mean convex boundary and nonnegative scalar curvature. We give an obstruction to filling 2-manifolds by such…

We provide an elementary proof of a result by V.P.~Fonf and C.~Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J.~Lindenstrauss and R.R.~Phelps \cite{LP} to…

Non-supersymmetric multi-wall configurations are generically unstable. It is proposed that the stabilization in compact space can be achieved by introducing a winding number into the model. A BPS-like bound is studied for the energy of…

We classify compact complex surfaces which contain a Zariski open subset whose universal covering is the cylinder DxC.

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property…

For manifolds with a distinguished asymptotically flat end, we prove a density theorem which produces harmonic asymptotics on the distinguished end, while allowing for points of incompleteness (or negative scalar curvature) away from this…

In this article, we will define non-commutative covering spaces using Hopf-Galois theory. We will look at basic properties of covering spaces that still hold for these non-commutative analogues. We will describe examples including coverings…

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 present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the…

As was recently observed by M. Xu and J. Wolf, there is a gap in Berard Bergery's classification of odd dimensional positively curved homogeneous spaces. Since this classification has been used in other papers as well, we give a modern,…

We prove that all extreme points of the unit ball of a Lipschitz-free space over a compact metric space have finite support. Combined with previous results, this completely characterizes extreme points and implies that all of them are also…

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature then $Y$ has a universal cover.

We discuss construction of coverings of the unit ball of a finite dimensional Banach space. The well known technique of comparing volumes gives upper and lower bounds on covering numbers. This technique does not provide a construction of…

We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic…

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) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this…

It is shown that if $C$ is a nonempty convex and weakly compact subset of a Banach space $X$ with $M(X)>1$ and $T:C\rightarrow C$ satisfies condition $(C)$ or is continuous and satisfies condition $(C_{\lambda})$ for some $\lambda \in…

We give a new characterization of spaces with nonnegative curvature in the sense of Alexandrov.