Nonnegative Ricci Curvature Research Articles

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Abstract 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 consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.

Bounds on volume growth of geodesic balls for Einstein warped products

The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped product. In particular, we present an obstruction for the existence of such a class of manifolds.

Splitting of 3-manifolds and rigidity of area-minimising surfaces

In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.

The rigidity theorem for harmonic maps from Berwald manifolds

In this paper, we can prove that any non-degenerate strongly harmonic map ϕ from a compact Berwald manifold with nonnegative general Ricci curvature to a Landsberg manifold with non-positive flag curvature must be totally geodesic, which generalizes the result of Eells and Sampson ([2]).

Li-Yau inequality on graphs

AbstractWe prove the Li-Yau gradient estimate for the heat kernel on graphs. The onlyassumption is a variant of the curvature-dimension inequality, which is purely local,and can be considered as a new notion of curvature for graphs. We compute thiscurvature for lattices and trees and conclude that it behaves more naturally than thealready existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth.We also derive Harnack inequalities and heat kernel bounds from the gradient esti-mate, and show how it can be used to strengthen the classical Buser inequality relatingthe spectral gap and the Cheeger constant of a graph. 1 Introduction and main ideas In their celebrated work [15] Li and Yau proved an upper bound on the gradient ofpositive solutions of the heat equation. In its simplest form, for an n-dimensional com-pact manifold with non-negative Ricci curvature the Li-Yau gradient estimate statesthat a positive solution uof the heat equation (∆−∂

On Ricci curvature and volume growth in dimension three

We prove that any complete metric on $\mathbb{R}^3$ minus an open ball, with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

The Liouville’s theorem of harmonic functions on Alexandrov spaces with nonnegative Ricci curvature

In this paper, the authors prove the Liouville’s theorem for harmonic function on Alexandrov spaces by heat kernel approach, which extends the Liouville’s theorem of harmonic function from Riemannian manifolds to Alexandrov spaces.

Rigidity in dimension four of area-minimising Einstein manifolds

AbstractThe aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand nonnegative Ricci curvature. Under a suitable choice, we have$area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.

Bose-Einstein condensation on a manifold with non-negative Ricci curvature

The Bose-Einstein condensation for an ideal Bose gas and for a dilute weakly interacting Bose gas in a manifold with non-negative Ricci curvature is investigated using the heat kernel and eigenvalue estimates of the Laplace operator. The main focus is on the nonrelativistic gas. However, special relativistic ideal gas is also discussed. The thermodynamic limit of the heat kernel and eigenvalue estimates is taken and the results are used to derive bounds for the depletion coefficient. In the case of a weakly interacting gas, Bogoliubov approximation is employed. The ground state is analyzed using heat kernel methods and finite size effects on the ground state energy are proposed. The justification of the c-number substitution on a manifold is given.

On Lorentzian manifolds with highest first Betti number

We consider Lorentzian manifolds with parallel light-like vector field V. Being parallel and light-like, the orthogonal complement of V induces a codimension one foliation. Assuming compactness of the leaves and non-negative Ricci curvature on the leaves it is known that the first Betti number is bounded by the dimension of the manifold or the leaves if the manifold is compact or non-compact, respectively. We prove in the case of the maximality of the first Betti number that every such Lorentzian manifold is – up to finite cover – diffeomorphic to the torus (in the compact case) or the product of the real line with a torus (in the non-compact case) and has very degenerate curvature, i.e. the curvature tensor induced on the leaves is light-like.

Topology of codimension-one foliations of nonnegative curvature. II

We prove that a 3-connected closed manifold of dimension does not admit a codimension-one -foliation of nonnegative curvature. In particular, this gives a complete answer to a question of Stuck on the existence of codimension-one foliations of nonnegative curvature on spheres. We also consider codimension-one -foliations of nonnegative Ricci curvature on a closed manifold with leaves having finitely generated fundamental group, and show that such a foliation is flat if and only if is a -manifold. Bibliography: 13 titles.

An Entropy Formula for the Heat Equation on Manifolds with Time-Dependent Metric, Application to Ancient Solutions

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed metrics with nonnegative Ricci curvature). As applications, we classify nonnegative ancient solutions to the heat equation according to their entropies. In particular, we show that a nonnegative ancient solution whose entropy grows sublinearly on a manifold evolving under super Ricci flow must be constant. The assumption is sharp in the sense that there do exist nonconstant positive eternal solutions whose entropies grow exactly linearly in time. Some other results are also obtained.

Two rigidity theorems for fully nonlinear equations

This paper is concerned with the fully nonlinear equation σ2(g)=aσ1(g)+b. The first result is to obtain the entire solutions of the equation for conformally flat metric on Rn under some additional assumptions, which generalizes the famous result of Chang–Gursky–Yang in [3]. The second one is to classify compact locally conformally flat manifolds with nonnegative Ricci curvature and the metric g satisfying the equation.

Sequences of Laplacian Cut-Off Functions

We derive several new applications of the concept of sequences of Laplacian cut-off functions on Riemannian manifolds (which we prove to exist on geodesically complete Riemannian manifolds with nonnegative Ricci curvature). In particular, we prove that this existence implies $$\mathsf {L}^q$$ -estimates of the gradient, a new density result of smooth compactly supported functions in Sobolev spaces on the whole $$\mathsf {L}^q$$ -scale, and a slightly weaker and slightly stronger variant of the conjecture of Braverman, Milatovic, and Shubin on the nonnegativity of $$\mathsf {L}^2$$ -solutions $$f$$ of $$(-\Delta +1)f\ge 0$$ . The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.

Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

In this paper, we give a rigidity theorem for a complete non-compact expanding (or steady) Ricci soliton with nonnegative Ricci curvature and certain scalar curvature decay condition. As an application, we prove that any complete non-compact expanding (or steady) Kähler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question proposed by Chow, Lu and Ni in case of Kähler-Ricci solitons.

Large time behavior of the heat kernel

In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding’s original strategy, by blowing down the metric, using Cheeger and Colding’s theory about limit spaces of Gromov-Hausdorff convergence, combining with the Gaussian upper bound of heat kernel on limit spaces, we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure. As one application, we get the consistent large time limit of heat kernel in more general context, which generalizes the former result of P. Li. Furthermore, by choosing different sequences to blow down the suitable metric, we show the first example manifold whose heat kernel has inconsistent limit behavior, which answers an open question posed by P. Li negatively.

Sequences of open Riemannian manifolds with boundary

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$\delta$ inner regions" which avoid the boundary by a distance $\delta$. We prove Gromov-Hausdorff compactness theorems for sequences of these "$\delta$ inner regions". We then build "glued limit spaces" out of the Gromov-Hausdorff limits of these $\delta$ interior regions and study the properties of these glued limit spaces. Our main applications assume the sequence is noncollapsing and has nonnegative Ricci curvature. We include open questions.

Stability properties and topology at infinity of $$f$$ f -minimal hypersurfaces

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the topology at infinity of $f$-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted $L^1$-Sobolev inequality for hypersurfaces in Cartan-Hadamard weighted manifolds, satisfying suitable restrictions on the weight function.

A Sharp Sobolev Interpolation Inequality on Finsler Manifolds

In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative Ricci curvature which supports the sharp Sobolev interpolation inequality is isometric to a Minkowski space. The proofs are based on properties of the Finsler–Laplace operator and on the Finslerian Bishop–Gromov volume comparison theorem.

Sharp Morrey-Sobolev Inequalities on Complete Riemannian Manifolds

Two Morrey-Sobolev inequalities (with support-bound and $L^1-$bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in $\mathbb R^n$. We prove the following results in both cases: $\bullet$ If $(M,g)$ is a {\it Cartan-Hadamard manifold} which verifies the $n-$dimensional Cartan-Hadamard conjecture, sharp Morrey-Sobolev inequalities hold on $(M,g)$. Moreover, extremals exist if and only if $(M,g)$ is isometric to the standard Euclidean space $(\mathbb R^n,e)$. $\bullet$ If $(M,g)$ has {\it non-negative Ricci curvature}, $(M,g)$ supports the sharp Morrey-Sobolev inequalities if and only if $(M,g)$ is isometric to $(\mathbb R^n,e)$.