共查询到20条相似文献,搜索用时 31 毫秒
1.
Jia Yong Wu 《数学学报(英文版)》2011,27(8):1591-1598
In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator (p ?? 2) along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma??s results [Ann. Glob. Anal. Geom., 29, 287?C292 (2006)]. 相似文献
2.
Forman 《Discrete and Computational Geometry》2003,29(3):323-374
Abstract. In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds.
We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions
to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing
the properties of the combinatorial Ricci curvature with those of its Riemannian avatar. 相似文献
3.
Forman 《Discrete and Computational Geometry》2008,29(3):323-374
Abstract. In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds.
We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions
to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing
the properties of the combinatorial Ricci curvature with those of its Riemannian avatar. 相似文献
4.
Liang Zhao 《Results in Mathematics》2013,63(3-4):937-948
In this paper, we derive the evolution equation for the eigenvalues of p-Laplace operator. Moreover, we show the following main results. Let ( ${M^{n}, g(t)), t\in [0,T),}$ be a solution of the unnormalized powers of the mth mean curvature flow on a closed manifold and λ1,p (t) be the first eigenvalue of the p-Laplace operator (p ≥ n). At the initial time t = 0, if H > 0, and $$h_{ij}\geq \varepsilon Hg_{ij}\quad \left(\frac{1}{p}\leq\epsilon\leq \frac{1}{n}\right),$$ then λ1,p (t) is nondecreasing and differentiable almost everywhere along the unnormalized powers of the mth mean curvature flow on [0,T). At last, we discuss some interesting monotonic quantities under unnormalized powers of the mth mean curvature flow. 相似文献
5.
Shahroud Azami 《复变函数与椭圆型方程》2020,65(5):775-784
ABSTRACTIn this article, we study the evolution, monotonicity for the first eigenvalue of the clamped plate on closed Riemannian manifold along the Ricci flow. We prove that the first nonzero eigenvalue is nondecreasing under the Ricci flow under certain geometric conditions and find some applications in 2-dimensional manifolds. 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.
Using Hamilton's Ricci flow we shall prove several pinching results for integral curvature. In particular, we show that if
p>n/2$ and the L
p
norm of the curvature tensor is small and the diameter is bounded, then the manifold is an infra-nilmanifold. We also obtain
a result on deforming metrics to positive sectional curvature.
Received: 17 February 1999 相似文献
8.
Weiyong He 《Journal of Geometric Analysis》2013,23(4):1876-1931
We show that Perelman’s ${\mathcal{W}}$ functional on Kähler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kähler–Ricci flow (the first Chern class is positive) can be generalized to Sasaki–Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectional curvature is preserved along Sasaki–Ricci flow, using Bando and Mok’s methods and results in Kähler–Ricci flow. In particular, we show that the Sasaki–Ricci flow converges to a Sasaki–Ricci soliton when the initial metric has nonnegative transverse bisectional curvature. 相似文献
9.
In this paper, we first derive a monotonicity formula for the first eigenvalue of on a closed surface with nonnegative scalar curvature under the (unnormalized) Ricci flow. We then derive a general evolution formula for the first eigenvalue under the normalized Ricci flow. As an application, we obtain various monotonicity formulae and estimates for the first eigenvalue on closed surfaces. 相似文献
10.
Ezequiel R. Barbosa Marcos Montenegro 《Bulletin of the Brazilian Mathematical Society》2008,39(3):427-445
In this work we present some properties satisfied by the second L
2-Riemannian Sobolev best constant along the Ricci flow on compact manifolds of dimensions n ≥ 4. We prove that, along the Ricci flow g(t), the second best constant B
0(2, g(t)) depends continuously on t and blows-up in finite time. In certain cases, the speed of the explosion is, at least, the same one of the curvature operator.
We also show that, on manifolds with positive curvature operator or pointwise 1/4-pinched curvature, one of the situations
holds: B
0(2, g(t)) converges to an explicit constant or extremal functions there exists for t large.
相似文献
11.
In this paper, we study the evolution of L
2 one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the L
2 norm of a smooth one form is non-increasing along the Ricci flow with bounded curvature. The L
∞ norm is showed to have monotonicity property too. Then we use L
∞ cohomology of one forms with compact support to study the singularity model for the Ricci flow on
. 相似文献
12.
Shu-Cheng Chang 《Geometriae Dedicata》1997,64(3):253-260
In this article, we prove the compactness of the set of critical 4-manifolds with L
p-bound on the negative part of the Ricci curvature tensor, p > 2. In earlier work we proved this under the assumption that the Ricci curvature is pointwise bounded from below. 相似文献
13.
Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L
2 harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L
2 harmonic spinors on spin manifolds. 相似文献
14.
Xiang Gao 《Journal of Mathematical Analysis and Applications》2012,387(2):791-798
In this paper, we establish a new curvature condition preserved by the Ricci flow, which is named as 2-parameters nonnegative curvature condition. It relies on the first, second and third eigenvalues of the Riemannian curvature operator. Based on this, we prove the strong maximum principle for the 2-parameters nonnegativity along Ricci flow. 相似文献
15.
16.
Laurent Veysseire 《Comptes Rendus Mathematique》2010,348(23-24):1319-1322
Most known lower bounds on the spectral gap of the Laplacian using Ricci curvature are based on the infimum of the Ricci curvature, and can be really poor when the Ricci curvature is large everywhere but on a small subset on which it is small. Here we show that the harmonic mean of the Ricci curvature is a lower bound on the spectral gap of the Laplacian, which partially solves the problem (unfortunately, we have to assume that the Ricci curvature is everywhere nonnegative). 相似文献
17.
In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M. 相似文献
18.
Zhongmin Qian 《Bulletin des Sciences Mathématiques》2009,133(2):145-168
In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L2-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations. 相似文献
19.
20.
We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures. 相似文献