Related papers: On the min-max width of unit volume three-spheres

We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3…

We construct a family of Riemannian 3-spheres that cannot be "swept out" by short closed curves. More precisely, for each $L > 0$ we construct a Riemannian 3-sphere $M$ with diameter and volume less than 1, so that every 2-parameter family…

We recall the definition of an r-maximal set in a metric space as a maximal subset of diameter r. In the special case when the metric space is Euclidean such a set is exactly a solid of constant diameter r. In the process of reviewing the…

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The…

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following…

Given $(M,g)$ a smooth compact Riemannian manifold without boundary of dimension $n\geq 3$, we consider the first conformal eigenvalue which is by definition the supremum of the first eigenvalue of the Laplacian among all metrics conformal…

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute…

We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at…

In this paper, inspired by Schur's comparison theorem about curves in Euclidean space, we mainly provide a Schur's type volume comparison theorem, which is about the volumes of the boundaries of open balls in a complete $n$-dimensional…

The present paper is a follow up of our paper \cite{nS}. We investigate here the maximization of higher order eigenvalues in a conformal class on a smooth compact boundaryless Riemannian surface. Contrary to the case of the first nontrivial…

On finite-volume hyperbolic $3$-manifolds, we compare volumes of different metrics using the exponential convergence of Ricci-DeTurck flow toward the hyperbolic metric $h_0$. We prove that among metrics with scalar curvature bounded below…

We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand,…

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…

In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend…

For $n\geq 25$, we construct a smooth metric $\tilde{g}$ on the standard $n$-dimensional sphere $\mathbb{S}^n$ such that there exists a sequence of smooth metrics $\{\tilde{g}_k\}_{k\in\mathbb{N}}$ conformal to $\tilde g$ where each $\tilde…

Let $M$ be an open manifold of dimension at least $3$, which admits a complete metric of positive scalar curvature. For a function $v$ with bounded growth of derivative, whether $M$ admits a metric of positive scalar curvature with volume…

Let \Sigma be a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of \Sigma is bounded from below by the volume of the unit ball in R^k. This answers a question posed by R.…

In this paper we prove the following. Let $\Sigma$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $\Sigma \times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian…

We give a sharp upper bound for the area of a minimal two-sphere in a three-manifold (M,g) with positive scalar curvature. If equality holds, we show that the universal cover of (M,g) is isometric to a cylinder.

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the…