Manifolds Of Nonnegative Ricci Curvature Research Articles

Maximal First Betti Number Rigidity for Open Manifolds of Nonnegative Ricci Curvature

Maximal First Betti Number Rigidity for Open Manifolds of Nonnegative Ricci Curvature

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Manifolds of nonnegative Ricci curvature and infinite topological type

Manifolds of nonnegative Ricci curvature and infinite topological type

5-dimensional space-periodic solutions of the static vacuum Einstein equations

An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S1 × S2 and sphere S3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.

Inequalities between Dirichlet, Neumann and buckling eigenvalues on Riemannian manifolds

In this paper, we discuss various comparison results between Laplacian (or bilaplacian) eigenvalues for a mean-convex bounded domain in a Riemannian manifold of non-negative Ricci curvature, under different boundary conditions, namely Dirichlet, Neumann, clamped and buckling conditions. We first obtain a new integral gradient estimate for a first Dirichlet eigenfunction, and deduce therefrom that the first buckling eigenvalue of the bi-Laplacian is lower than four times the first eigenvalue of the Dirichlet Laplacian. An immediate consequence is a new comparison result between Dirichlet and Neumann Laplacian first eigenvalues. These inequalities as well as the intermediate key results are given in the general context of weighted Riemannian manifolds and under various curvature conditions.

Couplings in $L^{p}$ distance of two Brownian motions and their Lévy area

Dans cet article, nous étudions les couplages co-adaptés de diffusions hypoelliptiques sur le groupe de Heisenberg (sous-riemannien). Ces diffusions canoniques, que nous appelons mouvement brownien sur le groupe de Heisenberg, sont constituées d’un mouvement brownien planaire et de son aire de Lévy. Nous prouvons que, contrairement au cas des variétés riemanniennes à courbure de Ricci positive ou nulle, pour tout couplage co-adapté, deux mouvements browniens partant de deux points donnés ne peuvent rester à distance bornée pour tout temps $t\geq 0$. Nous prouvons en fait le résultat plus fort qu’ils ne peuvent rester bornés dans $L^{p}$ pour $p\geq 2$. Nous établissons également deux résultats positifs. D’une part, nous étudions le couplage par réflexion et montrons qu’il reste borné dans $L^{p}$ pour $0<p<1$. D’autre part, nous construisons un couplage explicite statique (et en particulier non-adapté) entre les lois de deux mouvements browniens qui donne un contrôle en distance $L^{1}$ de Wasserstein uniforme en temps. Enfin, nous expliquons comment nos résultats se généralisent aux groupes de Heisenberg de dimension supérieure.

A Serrin-type symmetry result on model manifolds: An extension of the Weinberger argument

We consider the classical ‘‘Serrin's symmetry result” for the overdetermined boundary value problem related to the equation Δu=−1 in a model manifold of non-negative Ricci curvature. Using an extension of the Weinberger classical argument we prove a Euclidean symmetry result under a suitable ‘‘compatibility” assumption between the solution and the geometry of the model.

Mean curvature in manifolds with Ricci curvature bounded from below

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $\Sigma$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $\Sigma$ does not separate $M$ then $\Sigma$ is totally geodesic and $M\setminus\Sigma$ is isometric to the Riemannian product $\Sigma\times(a,b)$, and if $\Sigma$ separates $M$ then the map $i_*:\pi_1(\Sigma)\rightarrow \pi_1(M)$ induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature $H\geq(n-1)\sqrt{k}$ in a manifold of Ricci curvature $Ric_M\geq-(n-1)k,k>0$, and for a free boundary minimal hypersurface in a manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact $n$-dimensional manifold $N$ with the number of generators of $\pi_1(N)<n$ cannot be minimally embedded in the flat torus $T^{n+1}$.

Free boundary minimal annuli in convex three-manifolds

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds of nonnegative Ricci curvature. This includes strictly convex domains in $\mathbb{R}^3$, thereby solving an open problem of Jost.

Low energy canonical immersions into hyperbolic manifolds and standard spheres

We consider critical points of the global L2-norm of the second fundamental form, and of the mean curvature vector of isometric immersions of compact Riemannian manifolds into a fixed background Riemannian manifold, as functionals over the space of deformations of the immersion. We prove new gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the standard sphere can be cast as a theorem about their critical points having constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various minimal submanifolds that occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of the first of the functionals mentioned.

Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.

Minimal Graphic Functions on Manifolds of Nonnegative Ricci Curvature

We study minimal graphic functions on complete Riemannian manifolds ∑ with nonnegative Ricci curvature, euclidean volume growth, and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville-type theorem with such growth via splitting for tangent cones of ∑ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville-type theorem under a certain nonradial Ricci curvature decay condition on ∑. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section. © 2015 Wiley Periodicals, Inc.

Minimal Graphic Functions on Manifolds of Nonnegative Ricci Curvature

AbstractWe study minimal graphic functions on complete Riemannian manifolds ∑ with nonnegative Ricci curvature, euclidean volume growth, and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville‐type theorem with such growth via splitting for tangent cones of ∑ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville‐type theorem under a certain nonradial Ricci curvature decay condition on ∑. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section. © 2015 Wiley Periodicals, Inc.

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.

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Abstract We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger–Gromoll–Lichnerowicz Splitting Theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.

On Foxʼs m-dimensional category and theorems of Bochner type

We show that catm(X)=cat(jm), where catm(X) is Foxʼs m-dimensional category, jm:X→X[m] is the mth Postnikov section of X and cat(X) is the Lusternik–Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature.

On complete manifolds supporting a weighted Sobolev type inequality

This paper studies the geometric and topological properties of complete open Riemannian manifolds which support a weighted Sobolev or log-Sobolev inequality. We show that the constant in the weighted Sobolev inequality on a complete open Riemannian manifold should be bigger than or equal to the optimal one on the Euclidean space of the same dimension and that a complete open manifold of asymptotically non-negative Ricci curvature supporting a weighted Sobolev inequality must have large volume growth. We also show that a complete manifold of non-negative Ricci curvature on which the log-Sobolev inequality holds is not very far from the Euclidean space.

Spectrum of the Laplacian on quaternionic Kähler manifolds

Spectrum of the Laplacian on quaternionic Kähler manifolds

Manifolds with asymptotically nonnegative minimal radial curvature

Abstract In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by [U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4) 18 (1985), 651–670. MR839689 (87j:53058) Zbl 0595.53043], [A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. École Norm. Sup. (4) 21 (1988), 593–622. MR982335 (90d:53049) Zbl 0662.53032] and [S.-H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications. Amer. J. Math. 116 (1994), 669–682. MR1277451 (95c:53049) Zbl 0953.53027] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature.

The Caffarelli-Kohn-Nirenberg Inequalities on Complete Manifolds

We find a new sharp Caffarelli-Kohn-Nirenberg inequality and show that the Euclidean spaces are the only complete non-compact Riemannian manifolds of non-negative Ricci curvature satisfying this inequality. We also show that a complete open manifold with non-negative Ricci curvature in which the optimal Nash inequality holds is isometric to a Euclidean space