arXiv research

Researchers developed volume comparison theorems in Finsler spacetimes.

Paper improves volume comparisons and entropy estimates for integral Ricci curvature.

Volume comparison theorem for rank 1 symmetric spaces proved.

Paper proves a new volume comparison theorem for Riemannian manifolds.

Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.

The paper extends volume comparison results to total σ_l-curvature.

New volume comparison theorem for gradient Ricci almost solitons.

The paper proves estimates and theorems for Kähler manifolds.

The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.

The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.

Paper extends curvature estimates to new tensor types.

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…

Study on manifolds with density using modified Hessians for curvature comparison.

New proof of Bishop's theorem using soap bubbles with singularities.

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.

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

In this article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we g…

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

Study compares spectral properties of a specific tensor in geometry.

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…

Second part of Q-curvature research focusing on volume comparison.

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.

Study compares volumes of hyperbolic 3-manifolds using Ricci-DeTurck flow.

We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smoo…

In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$ and also the rigidity result when certain …

Bounding characteristic numbers of Riemannian manifolds via volume.

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.

On Kahler manifolds with Ricci curvature lower bound, assuming the real analyticity of the metric, we establish a sharp relative volume comparison theorem for small balls. The model spaces being compared to are complex space forms, i.e, Kahler manifolds with constant holomorphic sectional curvature. Moreover, we give a…

The paper extends a theorem about scalar curvature and volume in higher dimensions.

Paper proves new theorems about curvature in weighted manifolds.

In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.

The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.

In this paper, we study the complete bounded $λ$-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded $λ$-hypersurfaces with $|A|\leqα$ and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…

In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scala…

Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local v…

The study solves a problem in conformal geometry with applications to Q-curvature.

Uniform comparison of hyperbolic ball volumes on universal cover.

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to …

Study investor attention using search volume data before and after mobile device popularity.

The paper extends Schwarz's lemma to RC-positivity and complex manifolds.

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entit…

We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.

Extends tube volume estimates using integral curvature bounds.

Formulates Index III lemma and Rauch III theorem with applications.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

Gradient estimates for special harmonic functions on manifolds.