arXiv research

Upper bound on geodesic ball volume in Riemannian manifolds.

Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.

Round balls minimize liquid drop model volumes ≤ 1.

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g…

This paper confirms volumes of geodesic balls can identify 4D space forms.

Paper finds critical metrics with pinched curvature are geodesic balls.

In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…

New method uses relative capacities of geodesic balls to determine scalar curvature.

Uniform comparison of hyperbolic ball volumes on universal cover.

Smooth surface encloses less volume than a ball.

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…

Compactness theorem for manifolds with scalar curvature and entropy bounds.

Proves stability of cone-volume measure with nearly constant density.

If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $ε_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.

Study shows diffused interface flows to single diffused balls over time.

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifol…

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

Study $σ_2$-curvature and volume of compact manifolds, proving conditions for Einstein metrics and geodesic balls.

Constructs simplified or complexified simplicial complexes.

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifo…

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

New metric properties show volume constraints in collapsing spaces.

We prove a so called $κ$ non-inflating property for Ricci flow, which provides an upper bound for volume ratio of geodesic balls over Euclidean ones, under an upper bound for scalar curvature. This result can be regarded as the opposite statement of Perelman's $κ$ non-collapsing property for Ricci flow. These two resul…

Study cohomology of ball quotients and their compactifications.

The study provides volume growth estimates for specific types of manifolds.

Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.

We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in $\mathbb{C}^2$. When the hypersurface has a uniform global structure, we show that a metric ball of radius $δ\gg 1$ either has volume on the order of $δ^3$ or $δ^4$. We also give necessary and …

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

The paper classifies energy-minimizing sets in specific domains.

We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian ge…

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

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.

We have discovered a "little" gap in our proof of the sharp conjecture that in $\mathbb{R}^n$ with volume and perimeter densities $r^m$ and $r^k$, balls about the origin are uniquely isoperimetric if $0 < m \leq k - k/(n+k-1)$, that is, if they are stable (and $m > 0$). The implicit unjustified assumption is that the g…

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the…

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

Study on Funk geometry volume growth and polytope flags, verifying conjectures.

Study on curves on specific arithmetic quotients of hyperbolic 2-ball.

Two geodesic balls maximize the third Neumann eigenvalue in hyperbolic space.

Algorithm finds small confidence sets for arbitrary distributions.