Let $(N,g)$ be an $n$-dimensional complete Riemannian manifold with nonempty boundary $\pt N$. Assume that the Ricci curvature of $N$ has a negative lower bound $Ric\geq -(n-1)c^2$ for some $c>0$, and the mean curvature of the boundary $\pt N$ satisfies $H\geq (n-1)c_0>(n-1)c$ for some $c_0>c>0$. Then a known result (see \cite{LN}) says that $\sup_{x\in N}d(x,\pt N)\leq \frac 1c\coth^{-1}\frac{c_0}c$. In this paper, we prove that if the boundary $\pt N$ is compact, then the equality holds if and only if $N$ is isometric to the geodesic ball of radius $\frac 1c\coth^{-1}\frac{c_0}c$ in an $n$-dimensional hyperbolic space $\mathbb{H}^n(-c^2)$ of constant sectional curvature $-c^2$. Moreover, we also prove an analogous result for manifold with nonempty boundary and with $m$-Bakry-Émery Ricci curvature bounded below by a negative constant.
We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.
We prove that a Kahler manifold with positive bisectional curvature and maximal diameter is isometric to the complex projective space with the Fubini-Study metric.
In this paper we give a diameter bound for Sasaki manifolds with positive transverse Ricci curvature. As an application, we obtain the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure.
We define the orthogonal Bakry-Émery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on Kähler and quaternionic Kähler manifolds under positivity assumption on the orthogonal Bakry-Émery tensor. Moreover, under such assumptions on the orthogonal Bakry-Émery tensor and the holomorphic or quaternionic sectional curvature on a Kähler manifold or a quaternionic Kähler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.
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.
Let $M^n$ be a compact Kähler manifold with bisectional curvature bounded from below by $1$. If $diam(M) = π/ \sqrt{2}$ and $vol(M)> vol(\mathbb{C}\mathbb{P}^n)/ 2^n$, we prove that $M$ is biholomorphically isometric to $\mathbb{C}\mathbb{P}^n$ with the standard Fubini-Study metric.
This paper aims to study the $(m,ρ)$-quasi Einstein manifold. This article shows that a complete and connected Riemannian manifold under certain conditions becomes compact. Also, we have determined an upper bound of the diameter for such a manifold. It is also exhibited that the potential function acquiesces to the Hodge-de Rham potential up to a real constant in an $(m,ρ)$-quasi Einstein manifold. Later, some triviality and integral conditions are established for a non-compact complete $(m,ρ)$-quasi Einstein manifold having finite volume. Finally, it is proved that with some certain constrain…
Let $ρ_n(V)$ be the number of complete hyperbolic manifolds of dimension n with volume less than $V$. Burger, Gelander, Lubotzky, and Moses showed that when n>3 there exist a,b>0 depending on the dimension such that aV log(V) < log(ρ_n(V)) < bV log(V), for V >> 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially. Additionally, this bound holds in dimension 3.
In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (J. Differential Geom. 83, 377--405, 2009) and Limoncu (Math. Z. 271, 715--722, 2012). As an application, we shall give an upper diameter bound for compact Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient condit…
In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319--327, 2010), Futaki and Sano (Asian J. Math. 17, 17--32, 2013), and Futaki \textit{et al} (Ann. Global Anal. Geom. 44, 105--114, 2013).
In this article, we extend the compactness theorems proved by Sprouse and Hwang-Lee to a weighted manifold under the assumption that the weighted Ricci curvature is bounded below in terms of its weight function. With the help of the $\varepsilon$-range, we treat the case that the effective dimension is at most $1$ in addition to the case that the effective dimension is at least the dimension of the manifold. To show these theorems, we extend the segment inequality of Cheeger-Colding to a weighted manifold.
In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topology of compact manifolds with positive Ricci curvature. Moreover, we also obtain a weak Margulis's lemma for manifolds under a lower Ricci curvature bound.
The Chekanov theorem generalizes the classic Lyusternik-Shnirel'man and Morse theorems concerning critical points of a smooth function on a closed manifold. A Legendrian submanifold Λof space of 1-jets of the functions on a manifold M defines a multi-valued function whose graph is the projection of Λin J^0 M = M x R. The Chekanov theorem asserts that if Λis homotopic to the 1-jet of a smooth function in the class of embedded Legendrian manifolds, then such a graph of a multi-valued function must have a lot of points (their number is determined by the topology of M) at which the tangent plane t…
We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n} $ to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed curves, of all genus, in a compact locally conformally symplectic manifold.
In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then…
Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we estimate the diameter of $M$ in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539--546).
The standard Bonnet-Myers theorem says that if the Ricci scalar of a Riemannian manifold is bounded below by a positive number, then the manifold is compact. Moreover, a bound of its diameter is pointed out. The theorem was extended to Finsler manifolds. In this paper we prove that if a certain condition on the average of the Ricci scalar holds, then the Finsler manifold M is compact if the Ricci scalar is bounded above by the same positive number. An upper bound of the diameter is also found. With no condition on Ricci scalar itself but with a different one on its average, we find that the Fi…
Gromov showed that there is an upper bound on the Betti numbers of all closed Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether such manifolds (if simply connected) fall into only finitely many rational homotopy types. We give a negative answer, in fact in dimension 6, which is the smallest possible. We also give counterexamples to some related questions in dimensions 7 and 9, improving the original counterexamples by Fang and Rong which were in dimensions at least 22.
For a Poincare-Einstein manifold under certain restrictions, X. Chen, M. Lai and F. Wang proved a sharp inequality relating Yamabe invariants. We show that the inequality is true without any restriction.
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
We prove some sharp systolic inequalities for compact $3$-manifolds with boundary. They relate the (relative) homological systoles of the manifold to its scalar curvature and mean curvature of the boundary. In the equality case, the universal cover of the manifold is isometric to a cylinder over a disk of nonnegative constant curvature.
By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio-Kirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which we replace uniform compactness of the sequence with un…
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov--Lévy and Bakry--Le…
It is shown that certain diffeomorphism or homeomorphism groups with no restriction on support of an open manifold with finite number of ends are bounded. It follows that these groups are uniformly perfect. In order to characterize the boundedness several conditions on automorphism groups of an open manifold are introduced. In particular, it is shown that the commutator length diameter of the automorphism group $\mathcal D(M)$ of a portable manifold $M$ is estimated by $2fragd_{\mathcal D(M)}+2$, where $fragd_{\mathcal D(M)}$ is the diameter of $\mathcal D(M)$ in the fragmentation norm.
For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples we derive the axisymmetric Eule…
We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from [6] and [10].
This is the very first paper to focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.
In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative Ricci curvature that is sharp in the order of time parameter for the heat kernel on the Euclidean space.
In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.
Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler \cite{W:1} on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy \cite{P:1} in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to $\R^n$. Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $\R^n$. Comparing with the well known flatness results in \cite{An:1}, \cite{Ba:1} and \cite{BKN:1} on asymptotically flat manifolds and…
We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis of cylindrical regions ($\varepsilon$-tubes) under mean curvature flow. The results are new even in the most classical case of mean convex surfaces evolving by mean curvature flow in $\mathbb{R}^3$.
The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-reg…
We obtain a lower bound for the diameter of a solution to the Ricci flow on a compact manifold with nonvanishing first real cohomology. A consequence of our result is an affirmative answer to Hamilton's conjecture that a product metric on $(S^{1}\times S^{n-1}$ cannot arise as a final time limit flow.
In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds with boundary. We determine the limit spaces of inradius collapsed manifolds as Alexandrov spaces with curvature uniformly bounded below. When the limit space has co-dimension one, we completely determined the topology of inradius collapsed manifold in terms of singular $I$-bundle…
In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M^n is bounded from above by (n+2)!d/3, where d is the diameter of a manifold M^n. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold M^n is bounded from above by 2(n+2)!Fill Rad(M^n) and, as a corollary, by 2(n+2)!(n+1)n^n(n!)^{1/2}(vol(M^n))^{1/n}, where Fill Rad(M^n) …
A version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(1)Sp(1) component of the qc-Ricci curvature on a compact seven dimensional quaternionic contact manifold is established. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached if and only if the quaternionic contact manifold is a round 3-Sasakian sphere.
On a complete noncompact Kähler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by $m^2$ if the Ricci curvature is bounded from below by $-2(m+1)$. Then we show that if this upper bound is achieved then the manifold has at most two ends. These results improve previous results on this subject proved by P. Li and J. Wang in \cite {L-W3} and \cite{L-W} under assumptions on the bisectional curvature.
In this paper, we prove sharp gradient estimates for a positive solution to the heat equation $u_t=Δu+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the equation $u_t=Δu$ and $\log u$ is of sublinear growth in both spatial and time directions then $u$ must be constant. This gradient estimate is sharp since it is well-known that $u(x,t)=e^{x+t}$ satisfying $u_t=Δu$. We also emphasize that our results are better than those given by Jiang (\cite{XJ16}), Souplet-Zhang (\cite{SZ06}), Wu (\cite{Wu15, Wu17}), and others.
In this note we prove the following result: There is a positive constant $ε(n,Λ)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $Λ$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \geq -ε(n,Λ)$, then $M^n$ is diffeomorphic to the product $M_1\times ... \times M_k$, where each $M_i$ is either a complex projective space or an irreducible K$\ddot{a}$hler-Hermitian symmetric space of rank $\geq 2$. This resolves a conjecture of F. Fang under the additional upper bound restrictions on section…
In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the Kähler-Ricci flow and the Ricci flow.
