Related papers: The Penrose Inequality for Metrics with Singular S…
In this article we give a criterion for the existence of a metric of curvature $1$ on a $2$-sphere with $n$ conical singularities of prescribed angles $2\pi\vartheta_1,\dots,2\pi\vartheta_n$ and non-coaxial holonomy. Such a necessary and…
We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral $n$-varifolds $V$ in Riemannian manifolds which have both: (i) finite index on their smoothly embedded part; and (ii)…
We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for…
Consider a compact, orientable, three dimensional Riemannian manifold with boundary with nonnegative scalar curvature. Suppose its boundary is the disjoint union of two pieces: the horizon boundary and the outer boundary, where the horizon…
Two quantities quantifying uncertainty relations are examined. In J.Math.Phys. 48, 082103 (2007), Busch and Pearson investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall…
We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in space-time) and manifestly covariant…
The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of…
We investigate extremal metrics at which various types of rigidity theorems involving scalar curvatures hold. The rigidity we discuss here is related to the rigidity theorems presented by Mario Listing in his previous preprint. More…
One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e…
We establish curvature obstruction theorems for manifolds with boundary. Our main theorems show that, for dimensions up to 7, a topologically nontrivial compact manifold with boundary cannot have a metric of positive $m$-intermediate…
We prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal…
We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the…
The Moore-Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore-Penrose method that permits to deal with general relativity involving…
The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with non-negative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian…
In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the…
Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the…
There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of…
In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincar\'e type inequality. We apply such inequality to higher-order mean curvature of…
In this work, we present several rigidity results for compact free boundary hypersurfaces in initial data sets with boundary. Specifically, in the first part of the paper, we extend the local splitting theorems from [G. J. Galloway and H.…
In this short article, we establish a rigidity theorem for pairs of hyperquadrics in a weaker sense, i.e., we impose a condition that minimal rational curves are preserved, which is stronger than inheriting a sub-VMRT structure, a notion…