共查询到20条相似文献,搜索用时 0 毫秒
1.
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
2.
Wilderich Tuschmann 《Proceedings of the American Mathematical Society》2002,130(1):303-306
A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.
3.
Hongxin Guo 《Journal of Mathematical Analysis and Applications》2010,363(2):497-501
Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained. 相似文献
4.
David J. Wraith 《Annals of Global Analysis and Geometry》2007,32(4):343-360
We construct a new infinite family of Ricci positive manifolds, generalising a well-known result of Sha and Yang.
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We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature (PIC). Then we review an application of the classification theorem, which is from Chen and Zhu’s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature. 相似文献
7.
WANG PeiHe & WEN YuLiang School of Mathematical Sciences Qufu Normal University Qufu China 《中国科学 数学(英文版)》2011,(3)
Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n. 相似文献
8.
For any complete noncompact Kahler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow. 相似文献
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10.
Sharief Deshmukh 《Annali di Matematica Pura ed Applicata》2008,187(1):59-65
In this paper we study the role of constant vector fields on a Euclidean space R
n+p
in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R
n+p
with mean curvature vector field H and a constant vector field on R
n+p
, the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).
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11.
In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature. 相似文献
12.
Liang Ming Shen 《数学学报(英文版)》2015,31(9):1391-1414
In this paper, we study Ricci flow on noncompact 4-manifolds with uniformly positive isotropic curvature and with no essential imcompressible space form. That means there is positive lower bound of isotropic curvature and bounded geometry. Then by Perelman's technique, we can analyze the structures of such manifolds. 相似文献
13.
Changyu Xia 《Proceedings of the American Mathematical Society》1997,125(6):1801-1806
Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.
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We prove that there is a T 2-invariant Riemannian metric of positive Ricci curvature on every four-dimensional simply connected T 2-manifold. 相似文献
16.
For a compact Riemannian manifold M, we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space. 相似文献
17.
Ricardo Estrada 《Mathematical Methods in the Applied Sciences》2006,29(4):427-444
We show that any continuous plane path that turns to the left has a well‐defined distribution that corresponds to the radius of curvature of smooth paths. We show that the distributional radius of curvature determines the path uniquely except for a translation. We show that Dirac delta contributions in the radius of curvature correspond to facets, that is, flat sections of the path, and show how a path can be deformed into a facet by letting the radius of curvature approach a delta function. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
18.
A. Engoulatov 《Journal of Functional Analysis》2006,238(2):518-529
We derive a gradient estimate for the logarithm of the heat kernel on a Riemannian manifold with Ricci curvature bounded from below. The bound is universal in the sense that it depends only on the lower bound of Ricci curvature, dimension and diameter of the manifold. Imposing a more restrictive non-collapsing condition allows one to sharpen this estimate for the values of time parameter close to zero. 相似文献
19.
In this paper,we study steady Ricci solitons with a linear decay of sectional curvature.In particular,we give a complete classification of 3-dimensional steady Ricci solitons and 4-dimensional K-noncollapsed steady Ricci solitons with non-negative sectional curvature under the linear curvature decay. 相似文献
20.
In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature. 相似文献