arXiv research

Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.

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

Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.

The paper estimates the volume of singular points in evolving surfaces.

Improved volume estimates for right-angled polyhedra in hyperbolic space.

Estimates volume of convex Alexandrov spaces with boundary.

Estimates open sets for fibrations, leading to volume vanishing results.

Uniform volume estimate for Kähler metrics in big cohomology classes.

We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume.

Paper proves volume growth estimate for steady gradient Ricci solitons.

Uniform estimates for Kaehler metrics' diameters and volumes.

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

Study critical metrics on manifolds with boundary using integral and boundary estimates.

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 estimates area and volume for spacetimes with integral mean curvature bounds.

We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and giv…

The paper proves estimates and theorems for Kähler manifolds.

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a c…

We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.

SLS optimizes minimum-volume regions for conditional quantiles, bypassing density estimation.

In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of Kähler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local collapsing estimate and can be regarded as an analogue of the Bishop-Gromov volume compa…

In a remarkable article published in 1982, M. Gromov introduced the concept of minimal volume, namely, the minimal volume of a manifold $M^n$ is defined to be the greatest lower bound of the total volumes of $M^n$ with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value b…

This paper tackles traffic volume estimation challenges with a deep learning method.

This paper focuses on the problem of estimating historical traffic volumes between sparsely-located traffic sensors, which transportation agencies need to accurately compute statewide performance measures. To this end, the paper examines applications of vehicle probe data, automatic traffic recorder counts, and neural …

We obtain an estimate for the volume of neighbourhoods of sets of large curvature in three-dimensional Kähler-Einstein manifolds.

We prove a volume inequality for 3-manifolds having C^0 metrics "bent" along a hypersurface, and satisfying certain curvature pinching conditions. The result makes use of Perelman's work on Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of c…

Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.

Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.

We revisit and demonstrate the Epps effect using two well-known non-parametric covariance estimators; the Malliavin and Mancino (MM), and Hayashi and Yoshida (HY) estimators. We show the existence of the Epps effect in the top 10 stocks from the Johannesburg Stock Exchange (JSE) by various methods of aggregating Trade …

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

Given a hyperbolic 3-manifold M containing an embedded closed geodesic, we estimate the volume of a complete hyperbolic metric on the complement of the geodesic in terms of the geometry of M. As a corollary, we show that the smallest volume orientable hyperbolic 3-manifold has volume >.32 .

In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.

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.

Paper extends curvature estimates to new tensor types.

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…

This paper improves Green's function estimates for compact Kähler manifolds.

Proof shows volume equals integral points for certain manifolds.

Estimates lower bound for simplicial volume of certain manifolds.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also nee…

Improved lower bounds on 2-bridge link complexity.

Researchers find a way to estimate potential functions for quaternionic metrics.

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same $\mathbb{Z}/2$ homology class in $\mathbb{C}P^n$ (but not among all submanifolds in this…

Study shows complete affine manifolds have zero simplicial volume.

Gradient estimates for special harmonic functions on manifolds.

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …

Flat stable minimal hypersurfaces found in 6D space.