Related papers: Volume comparison via boundary distances

We prove that the boundary of an orbit space or more generally a leaf space of a singular Riemannian foliation is an Alexandrov space in its intrinsic metric, and that its lower curvature bound is that of the leaf space. A rigidity theorem…

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance.…

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the…

We address the question of whether a Riemannian manifold-with-boundary (M,g) in dimension two is uniquely determined from knowledge of the distances between points on its boundary. An affirmative answer is called boundary rigidity for…

In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from…

In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the metric space endowed with the…

We give a very short and rather elementary proof of Gromov's filling volume inequality for n-dimensional Lipschitz cycles (with integer and Z_2-coefficients) in $L^\infty$-spaces. This inequality is used in the proof of Gromov's systolic…

The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts…

We study a generalized boundary rigidity problem, which investigates whether the areas of embedded minimal surfaces can uniquely determine a Riemannian manifold with boundary. We prove that for a conformal perturbation of an analytic metric…

Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is…

We prove a Lipschitz-volume rigidity result for $1$-Lipschitz maps of non-zero degree between metric manifolds (metric spaces homeomorphic to a closed oriented manifold) and Riemannian manifolds. The proof is based on degree theory and…

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. {\color{blue}For a metric space, we define its boundary to be the completion of the space minus…

In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the…

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of…

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In…

This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also…

A compact metric surface $M$ isometrically fills a closed metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for every $x,y\in C=\partial M$; that is, $M$ does not introduce any ``shortcuts'' between points on its boundary. Gromov's…

This PhD dissertation covers a range of topics in Finsler geometry and Finsler gravity, most notably: (i) the characterization of Berwald spaces, (ii) pseudo-Riemann (non-)metrizability of Berwald spaces, (iii) $(\alpha,\beta)$-metrics,…

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with…

Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the…