相关论文: Bounded volume class and Cheeger isoperimetric con…

Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 which admit a Riemannian metric with sectional curvature bounded in absolute value by C and diameter uniformly…

We prove that an open manifold with nonnegative Ricci curvature, linear volume growth and noncollapsed ends always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets given large volumes…

We show that closed manifolds supporting a nonpositively curved metric with negative $([\frac{n}{4}]+1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov.

It is shown that if the Kato constant of the negative part of the Ricci curvature below a positive level is small, then the volume of the corresponding manifold can be bounded above in terms of the Kato constant and the total Ricci…

The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the…

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using…

We show that for any closed nonpositively curved Riemannian 4-manifold $M$ with vanishing Euler characteristic, the Ricci curvature must degenerate somewhere. Moreover, for each point $p\in M$, either the Ricci tensor degenerates or else…

In this short note, exploits of constructions of $\mathcal{F}$-structures coupled with technology developed by Cheeger-Gromov and Paternain-Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant…

We prove a lower bound for the Cheeger constant of a cylinder $\Omega\times (0,L)$, where $\Omega$ is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the…

Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group $\pi_1(M)$ is finitely generated. This result…

It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in K\"ahler…

We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighbourhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the…

In this paper, by employing the Gauss-Bonnet theorem for Riemannian simplices due to Allendoerfer and Weil, we show that if a closed nonpositively curved $4$-manifold has nonzero Euler characteristic, then its simplicial volume is…

Let (M,g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K of M, all volume preserving stable constant mean curvature surfaces of sufficiently large…

A recent result of Bader, Gelander and Sauer shows that for manifolds of pinched negative curvature, the torsion part of the homology can be controlled by the volume. This is done by constructing an efficient simplicial model of the thick…

Given $(\bar{M},\Omega)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos \theta$ is bounded away from zero, where…

The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{ess-Minvol}(M)$,…

In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the…

For every $d\ge 3$, we construct a noncompact smooth $d$-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below $1$. We construct a similar example also for the relative…

Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. Let $ (Z, \partial Z) $ be an oriented, compact manifold with (possibly empty) smooth boundary and $ \dim Z \geqslant 2 $. In this article, we show that if the…