Related papers: On Gromov-Hausdorff stability in a boundary rigidi…

The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape…

In this work, we consider static manifolds $M$ with nonempty boundary $\partial M$. In this case, we suppose that the potential $V$ also satisfies an overdetermined Robin type condition on $\partial M$. We prove a rigidity theorem for the…

The boundary rigidity problem is a classical question from Riemannian geometry: if $(M, g)$ is a Riemannian manifold with smooth boundary, is the geometry of $M$ determined up to isometry by the metric $d_g$ induced on the boundary…

We study the gain in regularity of the distance to the boundary of a domain in $\mathbb R^m$. In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the…

The Gromov-Hausdorff distance between two metric spaces measures how far the spaces are from being isometric. It has played an important and longstanding role in geometry and shape comparison. More recently, it has been discovered that the…

We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in…

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed…

Given metric quotients $S$ and $S_n$, $n \in \mathbb{N}$, of a metric space $X$, sufficient conditions are provided on the data defining them guaranteeing that $S$ is the Gromov-Hausdorff limit of $S_n$. These conditions are recognized…

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$,…

We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally…

We present theoretical properties of the space of metric pairs equipped with the Gromov--Hausdorff distance. First, we establish the classical metric separability and the geometric geodesicity of this space. Second, we prove an…

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the…

In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and…

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov…

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

We establish the Bonnet-Myers theorem and the Bishop-Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For $n\geq 3$, let $(M^n,g)$ be a simply connected compact smooth $n$-manifold with…

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…

We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal…

The purpose of this article is study rigidity of free boundary minimal two-disks that locally maximize the modified Hawking mass on a Riemannian three-manifold with positive lower bound on its scalar curvature and mean convex boundary.…

Let $M$ be a complete Riemannian manifold and $F\subset M$ a set with a nonempty interior. For every $x\in M$, let $D_x$ denote the function on $F\times F$ defined by $D_x(y,z)=d(x,y)-d(x,z)$ where $d$ is the geodesic distance in $M$. The…