共查询到20条相似文献,搜索用时 31 毫秒
1.
We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) as well as in terms of transportation inequalities for volume measures, heat kernels, and Brownian motions and in terms of gradient estimates for the heat semigroup. © 2004 Wiley Periodicals, Inc. 相似文献
2.
New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry–Émery Ricci curvature, the Escobar–Lichnerowicz–Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry–Émery Ricci curvature and the m-Bakry–Émery Ricci curvature bounded from below by a non-positive constant, the Li–Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results. 相似文献
3.
In terms of conditions on the curvature tensors of Riemann-Christoffel, Ricci, Weyl and Bochner we obtain several new characterizations of complex hyperspheres in complex projective spaces, of complex hypercylinders in complex Euclidean spaces and of complex hyperlanes in complex space forms.Aspirant N.F.W.O. (België). 相似文献
4.
In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space. 相似文献
5.
Jun Ling 《Annals of Global Analysis and Geometry》2007,31(4):385-408
We give new estimates on the lower bounds for the first closed and Neumann eigenvalues for compact manifolds with positive
Ricci curvature in terms of the diameter and the lower bound of Ricci curvature. The results sharpen the previous estimates.
相似文献
6.
An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the
full Riemannian curvature tensor. In this article, supposing (M
n
, g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that
L\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C
0 bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds.
Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M
n
× [0, T), the curvature tensor stays uniformly bounded on M
n
× [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented. 相似文献
7.
G. Santhanam 《Proceedings Mathematical Sciences》2008,118(4):569-572
Let M be a hypersurface in a simply connected space form . We prove some rigidity results for M in terms of lower bounds on the Ricci curvature of the hypersurface M. 相似文献
8.
In correspondence with the manifolds of quasi-constant sectional curvature defined (cf [5], [9]) in the Riemannian context,
we introduce in the K?hlerian framework the geometric notion of quasi-constant holomorphic sectional curvature. Some characterizations
and properties are given. We obtain necessary and sufficient conditions for these manifolds to be locally symmetric, Ricci
or Bochner flat, K?hler η-Einstein or K?hler-Einstein, etc. The characteristic classes are studied at the end and some examples are provided throughout.
相似文献
9.
Claude LeBrun 《Inventiones Mathematicae》2001,145(2):279-316
We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact
4-manifold with non-trivial Seiberg-Witten invariants. These allow one, for example, to exactly compute the infimum of the
L
2-norm of Ricci curvature for any complex surface of general type. We are also able to show that the standard metric on any
complex-hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature
plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results for Einstein metrics.
Oblatum 14-III-2000 & 8-II-2001?Published online: 4 May 2001 相似文献
10.
Dario Cordero-Erausquin Robert J. McCann Michael Schmuckenschläger 《Inventiones Mathematicae》2001,146(2):219-257
A concavity estimate is derived for interpolations between L
1(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp
and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these
theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The
method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal
mass transport and interpolating maps on a Riemannian manifold.
Oblatum 1-XII-2000 & 3-V-2001?Published online: 13 August 2001 相似文献
11.
In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)?−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)?−k, which generalizes a result of L. Ni (2004, 2006) [20] and [21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds. 相似文献
12.
Jong-Gug Yun 《Archiv der Mathematik》2005,85(2):167-174
In this paper, we give an upper bound on the growth of π1(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L1- sense.Received: 19 July 2004 相似文献
13.
Peter Imkeller 《Probability Theory and Related Fields》1991,89(3):261-283
Summary LetM be a martingale of pure jump type, i.e. the compensation of the process describing the total of the point jumps ofM in the plane.M can be uniformly approximated by martingales of bounded variation jumping only on finitely many axial parallel lines. Using this fact we prove a change of variables formula in which forC
4-functions f the processf(M) is described by integrals off
(k) (M),k=1, 2, with respect to stochastic integrators of the types expected: a martingale, two processes behaving as martingales in one direction and as processes of bounded variation in the other, and one process of bounded variation. Hereby we are led to investigate two types of random measures not considered so far in this context. By combination with the integrators already known, they might complete the set needed for a general transformation formula. 相似文献
14.
Sharp estimates for the Ricci curvature of a submanifold M
n
of an arbitrary Riemannian manifold N
n+p
are established. It is shown that the equality in the lower estimate of the Ricci curvature of M
n
in a space form N
n+p
(c) is achieved only when M
n
is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in N
n+p
(c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.
相似文献
15.
In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery
Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula
for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions.
In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact
Riemannian manifolds. 相似文献
16.
Qi-hua Ruan 《Potential Analysis》2006,25(4):399-406
In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L
2– estimates of scalar curvature.Project partially supported by Yumiao Fund of Putian University. 相似文献
17.
In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L
n
+1 as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike
hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L
n
+1 which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates
for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in
L
n
+1 which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and
of a generalization of it.
Received: 5 July 1999 相似文献
18.
We establish dimension-independent estimates related to heat operators e tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied. 相似文献
19.
YONGFAN Zheng 《Geometriae Dedicata》1997,67(3):295-300
Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere. 相似文献
20.
Changyu Xia 《Monatshefte für Mathematik》1999,128(2):165-171
Let M be an n-dimensional simply connected Hadamard manifold with Ricci curvature satisfying and be a bounded domain having smooth boundary. In this paper, we prove that the first n nonzero Neumann eigenvalues of the Laplacian on Ω satisfy , where is a computable constant depending only on and , Ω being the volume of Ω. This result generalizes the corresponding estimate for bounded domains in a Euclidean space obtained
recently by M. S. Ashbaugh and R. D. Benguria.
(Received 19 May 1998; in revised form 21 September 1998) 相似文献