The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces
Minoru Tanaka, Kei Kondo
备注：
Change in title. This v3 is a version to appear in Nagoya Mathematical Journal. 10 pages, no figures
发表信息：
Nagoya Mathematical Journal, vol. 209 (2013) 23-34
We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we will prove that a complete non-compact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary, if the manifold M is not less curved than a non-compact model surface of revolution, and if the total curvature of the model surface is finite and less than $2π$. Hence, in the first result mentioned above, we may treat a much wider class of metrics than that of a complete non-compact Riemannian manifold whose sectional curvature is bounded from below by a constant.
This paper introduces and investigates a generalization of the notion of a pointed Riemannian manifold having its radial curvature bounded from above by that of a model surface of revolution.
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n under some conditions on the density of rays starting from the base point p or on the volume growth of geodesic balls in M.
We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of prolate type, the diameter of $M$ does not exceed the diameter of the ellipsoid, and if the diameter of $M$ equals that of the ellipsoid, then $M$ is isometric to the $n$-dimensional ellipsoid of revolution.
We show that a closed orientable Riemannian $n$-manifold, $n \ge 5$, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of $S^{n-1} \times S^1$.
In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.
We establish two comparison results between the solutions of a class of mean curvature equations and pieces of arcs of circles that satisfy the same Neumann boundary condition. Finally we present a number of examples where our estimates can be applied, some of them have a physical motivation.
The study of radial prescribing scalar curvature of X.Xu and P.C.Yang [2] in 1993 showed a nonexistence result on $S^2$. Later in 1995, W.Chen and C.Li [2] generalized the nonexistence result to higher dimensions. G.Bianchi and E.Egnell [1] suggested that there may exist some non-negative smooth radial function which cannot be scalar curvature in the standard conformal class of $RP^n.$ However, in our paper, we prove that all smooth radial non-negative smooth functions which are positive on the pole could be prescribing scalar curvatures. We consider the quotient $\frac{v_λ}{V_λ}$ rather than …
We survey all results concerning the topology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature that have no additional conditions other than restrictions to the dimension, volume growth or diameter growth of the manifold. We will also present relevant examples and list open problems.
We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than $π$. We show, as our main theorem, that all Busemann functions on $M$ are exhaustions, and that there exists a compact subset of $M$ such that the compact set contains all critical points for any Busemann function on $M$. As corollaries by the main theorem, $M$ has finite topological type, and the isometry group of $M$ is comp…
In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative $N$-Bakry Émery Ricci Curvature. In addition, we show that if $M^n$ is a complete, noncompact Riemannian manifold with nonnegative $N$-Bakry Émery Ricci curvature where $N>n$, then $H_{n-1}(M,\mathbb{Z})$ is $0$.
A gap in the proof prevents us to show that surfaces with constant mean curvature closed to 1/2 in H2 X R and having boundary with curvature greater than one, contained in a horizontal section P of H2 X R are topological disks, provided they are contained in one of the two halfspaces determined by P. This is the analogue in H2 X R of a result in R3 by A. Ros and H. Rosenberg [13, Theorem 2].
The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and it provides a tool using geometric flow to find such hypersurfaces with constant mean curvature in $\mathbb{H}^{n+1}$. Similar to the usual mean curvature flow, the MMCF is the natural negative $L^2$-gradient flow of the area-volume functional $\mathcal{I}(Σ)=A(Σ)+σV(Σ)$ associa…
Let $\mathcal{C}$ be the space of smooth metrics $g$ on a given compact manifold $M^{n}$ ($n\geq3$) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space $\mathcal{C}$ (we shall refer to this critical point as CPE metrics) under assumption that $(M,g)$ has zero radial Weyl curvature. Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard $3$-sphere. We will also p…
We define a Gauss map $γ:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $γ$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $γ$, are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of $\mathbb{CP}^{3}$ and obtain similar results for CMC hypersurfaces of this space.
We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article, as its applications, we prove the finiteness of topological type and a diffeomorphism theorem to Euclidean spaces.
We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $M$ with a base point $p \in M$ whose radial curvature at $p$ is bounded from below by that of a non-compact model surface of revolution $\tilde{M}$ which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point $\tilde{p} \in \tilde{M}$. Notice that our model $\tilde{M}$ does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove …
Dedicated to Professor K. Shiohama on the occasion of his seventieth birthday: This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold $M$. In the first series [arXiv:0901.4010], we showed that all Busemann functions on an $M$ which is not less curved than a von Mangoldt surface of revolution are exhaustions, if the total curvature of the surface is greater than $π$. A von Mangoldt surface of revolution is, by definition, a complete surface of revolution homeomorphic to Euclidean plane whose Gaussian curvature is non-increasing along e…
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università di Napoli Federico II, Italy（那不勒斯费德里科二世大学数学与应用系）；Scuola Superiore Meridionale, Napoli, Italy（那不勒斯南方高等学院）
In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.
In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with st…
We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some topological applications are discussed. We also give examples of manifolds of positive bi-Ricci curvature and prove that the connect sum of manifolds of positive bi-Ricci curvature admits metrics of positive bi-Ricci curvature.
Oberwolfach Workshop Report: Analysis, Geometry and Topology of Positive Scalar Curvature Metrcs: Limits of sequences of manifolds with nonnegative scalar curvature and other hypotheses
Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumptions on the warp potential.
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Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question has a non-trivial Seiberg-Witten invariant. However, it has recently been discovered that similar statements also apply to other parts of the curvature tensor. This article presents the most salient aspects of these curvature estimates in a self-contained manner, and shows how they can be applied to the theory of Einstein manifolds. We then probe the issue o…
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The aim of this article is to establish a Toponogov type triangle comparison theorem for Finsler manifolds, in the manner of radial curvature geometry. We consider the situation that the radial flag curvature is bounded below by the radial curvature function of a non-compact surface of revolution, the edge opposite to the base point is contained in a Berwald-like region, and that the Finsler metric is convex enough in the radial directions in that region.
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