arXiv research

Euclidean volumes of hyperbolic knots are algebraic numbers.

Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.

Upper bound on geodesic ball volume in Riemannian manifolds.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

The paper proves conditions for isoperimetric regions in curved spaces.

The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the E…

Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Study shows expanding Ricci solitons from specific metric cones.

Compactness theorem for manifolds with scalar curvature and entropy bounds.

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In…

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

Formula derived for volume entropy of certain metrics on Euclidean space.

The simplicial volume of non-R^3 contractible 3-manifolds is infinite.

Survey on 4-manifolds with specific curvature properties.

Flat space for manifolds with tiny curvature.

Hexagonal tilings minimize perimeter with unequal volumes.

Study on minimal submanifolds with finite curvature in Euclidean space.

The paper proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …

Proves inequalities on curved spaces with positive curvature.

Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.

In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…

In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

Study shows spheres in high dimensions have maximum volume if they are smooth and have a specific reach.

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one…

Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic vol…

Weyl's tube formula holds for various cross-sections under symmetry conditions.

The study classifies complete self-shrinkers in Euclidean space.

We present in this paper a \boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.

Let $Σ$ be a $k$-dimensional complete proper minimal submanifold in the Poincaré ball model $B^n$ of hyperbolic geometry. If we consider $Σ$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Σ$ and the ideal boundary $\partial_\infty Σ$, say $\…

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…

The paper examines properties and rigidity of self-expanders in Euclidean space.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

A 3-manifold's Ricci pinching condition implies it's flat if it has Euclidean volume growth.

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example…

Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…

Paper proves a new volume comparison theorem for Riemannian manifolds.

The study proves uniqueness of large isoperimetric sets in specific noncompact manifolds.

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

Formula calculates volume of CMC surfaces with translational periods, disproving isoperimetric conjecture.

In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.