arXiv research

New theorem shows curvature concentration depends linearly on volume ratio.

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

The paper studies the past inextendibility of FLRW spacetimes using the VDR asymptote.

Sharp Sobolev and Michael-Simon inequalities on curved manifolds.

The paper establishes criteria for spacetime inextendibility using asymptotic volume-distance-ratio analysis.

We study the asymptotic volume ratio of non-steady gradient Ricci solitons. Moreover, a local estimate of the volume ratio is obtained for expanding solitons which satisfy $\lim_{dist(O,x)\rightarrow\infty} |Sect|\cdot dist(O,x)^2=0$. Therefore, for such a soliton, we can show that it must have $\mathbb{R}^n$ as one of…

The study finds a limit on the volume growth of certain 3-manifolds.

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

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.

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold $(M^3, g)$ with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermo…

The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.

Estimates on Einstein manifolds improve Brownian motion behavior and curvature limits.

Infinite hyperbolic manifolds share same perimeter-to-volume ratio.

A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic volume ratio of gradient Ricci solitons.

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a…

Paper uses Transformers to predict intraday volume ratio with high accuracy.

The paper proves a Willmore-type inequality for unbounded convex sets.

For a compact right-angled polyhedron $R$ in $\mathbb H^3$ denote by $\operatorname{vol} (R)$ the volume and by $\operatorname{vert} (R)$ the number of vertices. Upper and lower bounds for $\operatorname{vol} (R)$ in terms of $\operatorname{vert} (R)$ were obtained in \cite{A09}. Constructing a 2-parameter family of po…

In this short note we observe that a recent result of C.-W. Chen meshes well with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L. Ni, and S.-J. Zhang to give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci solitons to have positive asymptotic volume ratio.

The paper calculates the asymptotics of quantum invariants for Whitehead chains.

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that $h_{d}(M)$ is finite. Later on, there are many researches which give better estimate…

Study shows robust method for estimating density ratios even with heavy contamination.

Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.

Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, whe…

Geometrically proves majorizing measure theorem on Hadamard manifolds.

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We…

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

Paper proves inequality for p-Laplacian eigenvalues on curved spaces.

Study shows eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

In this paper, we prove that any non-flat ancient solution to Kähler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier resul…

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume |M| of M is equal to vol(M)/v_n, where v_n is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while J…

The study proves surfaces with high genus have a specific inequality.

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

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…

Defines renormalized volume for bounded regions in asymptotically hyperbolic Einstein spaces.

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…

Study curvature and torsion from cross-ratios in discrete curves.

Study shows volume density in central harmonic spaces can vary arbitrarily.

Estimates for plate eigenvalues with nonzero Poisson's ratio.

Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.

The paper connects mass, harmonic functions, and capacity in asymptotically flat 3-manifolds.