arXiv research

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…

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

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.

Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volum…

Researchers developed volume comparison theorems in Finsler spacetimes.

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.

Sharp gradient estimates for positive Ricci curvature manifolds.

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…

Paper proves new theorems about curvature in weighted manifolds.

Bray's football theorem (\cite{bray2009penrose}) is a weakening of Bishop theorem in dimension 3. It gives a sharp volume upper bound for a three dimensional manifold with scalar curvature larger than $n(n-1)$ and Ricci curvature larger than $\varepsilon$. This paper extends Bray's football theorem in high dimensions, …

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…

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

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we o…

The study establishes comparison theorems for weighted Finsler manifolds and spacetimes.

Enhanced Bishop-Gromov theorem for homogeneous and inhomogeneous spaces.

This note explores comparison geometry concepts and theorems.

On Kahler manifolds with Ricci curvature bounded from below, we establish some theorems which are counterparts of some classical theorems in Riemannian geometry, for example, Bishop-Gromov's relative volume comparison, Bonnet-Meyers theorem, and Yau's gradient estimate for positive harmonic functions. The tool is a Boc…

The study compares spectral volumes of manifolds with weakly convex boundaries.

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

Enhances the Bishop-Gromov theorem for curved spaces, especially at late times.

For a complete Riemannian manifold $M$ with an (1,1)-elliptic Codazzi self-adjoint tensor field $A$ on it, we use the divergence type operator ${L_A}(u): = div(A\nabla u)$ and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems like mean curvature…

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

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…

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

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…

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

Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.

The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.

The paper explores curvature-free effects in manifolds with volume growth and ends-counting.

Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…

Sharp bounds and rigidity theorems for eigenvalues on manifolds.

Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume compari…

Entropy derived from Colding's volume on Ricci-flat manifolds.

We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the cen…

Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.

The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

Paper proves a new volume comparison theorem for Riemannian manifolds.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

Study compares nonsmooth spaces with integrable Ricci bounds.

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery's $CD(K,n)$ condition for linear graphs and prove the triviality of edge w…

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

Volume comparison theorem for rank 1 symmetric spaces proved.

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

The paper proves estimates and theorems for Kähler manifolds.

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

New volume comparison theorem for gradient Ricci almost solitons.

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.