If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g) contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n-space.
If $(M^n, g)$ is a closed Riemannian manifold where every unit ball has volume at most $ε_n$ (a sufficiently small constant), then the $(n-1)$-dimensional Uryson width of $(M^n, g)$ is at most 1.
We prove global estimates for the sub-Riemannian distance of CR Sasakian manifolds with non negative horizontal Webster-Tanaka Ricci curvature. In particular, in this setting, large sub-Riemannian balls are comparable to Riemannian balls.
In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermit…
A well-known and interesting family of sub-Riemannian space are the systems involving two balls rolling against each other without slipping or twisting. In this note, we show how the sub-Riemannian geodesics of these space, when the two balls are embedded in $\mathbb{R}^3 \times \mathbb{R}^3$, are horizontal curves on the intersections of these balls with Euclidean 5-planes.
We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifold, there is a unique family concentric pseudo balls which contains all the pseudo balls $C^{2,α}$-close to small spheres. This allows us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.
We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure
The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow $g(t)$. The considered flow in covariant symmetric $2$-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of $g(t)$. Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most handled situation we express…
Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local version of the Gunther and Bishop theorems is given and applied.
In this paper, we will count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.
We study the local Szegö-Weinberger profile in a geodesic ball $B_g(y_0,r_0)$ centered at a point $y_0$ in a Riemannian manifold $(\M,g)$. This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $μ_2$ of the Laplace-Beltrami Operator $Δ_g$ on $\M$ among subdomains of $B_g(y_0,r_0)$ with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of $\M$ at $y_0$. As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the …
In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.
For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, pa…
In this note, we prove two Kazdan-Warner type identities involving $v^{(2k)}$, the renormalized volume coefficients of a Riemannian manifold $(M^n,g)$, and $G_{2r}$, the so-called Gauss-Bonnet curvature, and a conformal Killing vector field on $(M^n,g)$. In the case when the Riemannian manifold is locally conformally flat, $v^{(2k)}=(-2)^{-k}σ_k$, $G_{2r}(g)=\frac{4^r(n-r)!r!}{(n-2r)!}σ_r$, and our results reduce to earlier ones established by Viaclovsky and by the second author.
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…
In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $Σ$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $Σ$. Furthermore, if $Σ$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $Σ$ is isometric to $\mathbb{S}^4$ up to scaling and $M$ splits in a neighborhood of $Σ$. Also, we obtain a rigidity result for the Riemannian…
We show that the volume of a simple Riemannian metric on $D^n$ is locally monotone with respect to its boundary distance function. Namely if $g$ is a simple metric on $D^n$ and $g'$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g')\ge vol(D^n,g)$. Furthermore, the same holds for Finsler metrics and the Holmes--Thompson definition of volume. As an application, we give a new proof of the injectivity of the geodesic ray transform for a simple Finsler metric.
Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem asserts that $g_0$ minimizes a normalized volume growth entropy among all complete, finite-volume, Riemannian metric on $M$. We will repair a gap in the proof when $g_0$ is locally isometric to the Cayley hyperbolic space.
We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary $C^2$ surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable $C^2$ surface $Σ$ in the Heisenberg group $\mathbb{H}^1$ and the sub-Riemannian sphere $\mathbb{S}^3$ of constant curvature $1$. In $\mathbb{H}^1$ we conclude that $Σ$ is a E…
Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. I…
We investigate the effect of the average scalar curvature on the conjugate radius, average area of the geodesic spheres, average volume of the metric balls and the total volume of a closed Riemannian manifold $N$ (or more generally $N$ with finite volume whose negative Ricci curvature integral on $SN$ is finite). For example, we prove that if the average scalar curvature is larger than the lower bound of the normalized Ricci curvature, then we can improve the Bishop-Gromov estimate on the average volume of the metric balls of any size. We also prove the monotone decreasing property of a certai…
We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth condition.
Given a compact manifold $M$ and a Riemannian manifold $N$ of bounded geometry, we consider the manifold ${\rm Imm} (M,N)$ of immersions from $M$ to $N$ and its subset ${\rm Imm}_μ(M,N)$ of those immersions with the property that the volume-form of the pull-back metric equals $μ$. We first show that the non-minimal elements of ${\rm Imm}_μ(M,N) $ form a splitting submanifold. On this submanifold we consider the Levi-Civita connection for various natural Sobolev metrics write down the geodesic equation and show local well-posedness in many cases. The question is a natural generalization of the …
We show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type.
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.
For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, unde…
In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.
In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar curvature and surface area of a geodesic ball in a Riemannian manifold with a pole satisfying steady Yamabe soliton.
We prove the diameter of the intersection of two closed convex balls in a Riemannian manifold eventually decreases continuously as the centers of the balls move apart.
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this ball.
We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci curvature lower bound $(n-1)k$, the geodesic ball of radius $l$ in the space form of curvature $k$ has the largest area-to-volume ratio. A similar but reversed inequality holds if we replace a lower bound on the cut distance by a lower bound of the mean curvature. As an application we show that $C^2$ isoperimetric domains in standard space forms are balls. Gene…
We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on…
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riem…
For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then the realizable forms as stable ball of Riemannan manifolds of dimension larger than three. For a Riemannian manifold $(M, g)$ fixed, we show that a broad class of polytopes can appear as stable ball of metrics in the conformal class of $g$. We use for that a polyhedral technique.
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.
We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. We provide exponential decay estimates for the biharmonic heat kernel under assumptions on the lower bound of Ricci curvature and noncollapsing of unit balls. And we prove a uniqueness criteria for the Cauchy problem. As corollaries we prove the conservation law for the biharmonic heat kernel and a uniform L-infinite estimate for entire solutions starting with bounded initial data.
We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjecture. We also prove a version with boundary. Kleiner and Lott independently proved similar results.
In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes don't involve noncommutative geometry or spin structures at all.
We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with strong bounded geometry, as in Definition $2.3$, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_0\in\mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of $\mathbb{M}^n_{k_0}$ the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of [Dru02b] …
In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet $p$-Laplacian ($1<p<\infty$) obtained by Matei [A.-M. Matei, First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the $p$-Laplacian in a Riemannian manifold, Tokyo J. Math. 21 (1998) 135--140], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet $p$-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricc…
Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close and almost isometric to a geodesic sphere $S(p_0,R_0)$ in $N$, but also its enclosed domain is $C^{1,α}$-close to a geodesic ball of constant curvature.
Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-Hölder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.
We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the {\it original manifold}. These results are different from most well known ones in the literature where the conclusions are made on th…
In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in \cite{BG1} and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of \cite{BG1}.
For positive $p$-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension $n$, $p$ and the radius of the ball on which the function is defined. Our approach is based on a careful application of the Moser iteration technique and is different from Cheng-Yau's method employed by Kostchwar and Ni, in which a gradient estimate for positive $p$-harmonic functions is derived under the assumption that the sectional curvature is bounded from below.
In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $δ>0$ such that if $(M,hyp)$ is a closed hyperbolic surface and $h$ another metric on $M$ with $\area(M,h)\leq δ\area(M,hyp)$ then for every radius $R\geq 1$ the universal cover of $(M,h)$ contains an $R$-ball with area at least the area of an $R$-ball in the hyperbolic plane. This positively answers a question of L. Guth for surfaces. We also prove an analog theorem for graphs.
