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 compact.
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.
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 $0$.
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 wide…
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.
In this work we study surfaces in radial conformally flat spaces. We characterize surfaces of rotation with constant Gaussian and Extrinsic curvature in these radial 3-spaces. We prove that all the spheres in the conformal 3-space have constant Gaussian curvature $K=1$ if, and only if, the conformal factor is special. In this special case we study geometric properties of this ambient 3-space, and as an application we prove that it is isometric to the space ${\mathbb{S}}^2\times {\mathbb{R}}$, so we consider it as the {\em Radial Model} of ${\mathbb{S}}^2\times {\mathbb{R}}$. We obtain two clas…
We prove the existence and uniqueness of radial graphs over a given domain of $\mathbb{S}^{n}$ having boundary on the sphere $\mathbb{S}^{n}$ and whose mean curvature at every point equals a prescribed positive function satisfying suitable barrier-type and monotonicity conditions.
An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik-Schnirelmann category. This part extends substantially the results of Q.M.Wang in \cite{Wa88}. The second part is concerned with their curvatures, more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induc…
In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $Ω\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar c…
Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.
In this paper we solve the Björling problem for the class of immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in $\mathbb{R}^3$
We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r)σ$ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over $S^n$ the flow exists for all times and remains a graph over $S^{n}$. Under weak further assumptions on the ambient manifold, we prove optimal decay of the gradient and that the flow leaves become umbilic exponentially fast. We prove optimal $C^2$ estimates in case that the ambient pinching improves.
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…
We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups they are still diffeomorphic to connected sums of spherical space forms and spherical bundles over the circle.
In this paper we shall establish that properly embedded constant mean curvature one surfaces in H^3 of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this nature. In general each annular end of such a surface is asymptotic to an end of a horosphere or an end of a catenoid cousin.
We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.
We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate that the norm of the second fundamental form decays to zero at infinity. The class of submanifolds $M$ with $a(M)<1$ contains all complete minimal surfaces in $\mathbb{R}^{n}$ with finite total curvature, all $m$-dimensional minimal submanifolds $M $ of $ \mathbb{R}^{n}$ with finite total scalar curvature $\smallint_{M}| α|^{m} dV<\infty $ and all complete 2-dimensional com…
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.
Let $M^n$ be a complete, open Riemannian manifold with $\Ric \geq 0$. In 1994, Grigori Perelman showed that there exists a constant $δ_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $α_M := \lim_{r \to \infty} \frac{\Vol(B_p(r))}{ω_n r^n} \geq 1-δ_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $α(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.
The regularity of limit spaces of Riemannian manifolds with L^p curvature bounds, $p > n/2$, is investigated under no apriori non-collapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with $L^p$ curvature bounds and an a priori volume growth assumption in the pointed Cheeger-Gromov topology. A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger-Gromov topo…
The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature $K_{\rm rad.}$ of an end converges to a constant -1 at infinity with the decay order $K_{\rm rad.} + 1 = o(r^{-1})$ and prove the absence of eigenvalues embedded in the essential spectrum. Furthe…
We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.
We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.
Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on …
In this paper we discuss and prove $ε$-regularity theorems for Einstein manifolds $(M^n,g)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If $x\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\ell},y^*)\subset \mathbb{R}^{n-\ell}\times C(Y^{\ell-1})$ for $\ell\leq 3$, then in fact $|Rm|\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting, and in fact it is easy to see wi…
