共查询到20条相似文献,搜索用时 479 毫秒
1.
Bing Ye Wu 《Geometriae Dedicata》2013,162(1):337-344
In 1968 Milnor conjectured that the fundamental group of any complete Riemannian manifold with nonnegative Ricci curvature is finitely generated. In this paper we obtain two results concerning Milnor’s conjecture. We first prove that the generators of fundamental group can be chosen so that it has at most logarithmic growth. Secondly we prove that the conjecture is true if additional the volume growth satisfies certain condition. 相似文献
2.
本文研究了具有非负Ricci曲率和次大体积增长的完备黎曼流形的拓扑结构问题.利用Toponogov型比较定理及临界点理论,获得了流形具有有限拓扑型的结果,推广了H.Zhan和Z.Shen的定理,并且还证明了该流形的基本群是有限生成的. 相似文献
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4.
In this paper, we can prove that any non‐degenerate strongly harmonic map ? from a compact Berwald manifold with nonnegative general Ricci curvature to a Landsberg manifold with non‐positive flag curvature must be totally geodesic, which generalizes the result of Eells and Sampson ([2]). 相似文献
5.
We say that a nonnegatively curved manifold (M, g) has quarter-pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature
is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem
of Brendle and Schoen it follows that any positively curved manifold with strictly quarter-pinched flag curvature must be
a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd-dimensional manifolds we obtain
results for the case that the flag curvature is pinched with some constant below one quarter, one of which generalizes a recent
work of Petersen and Tao. 相似文献
6.
This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting. 相似文献
7.
In this paper we address the issue of uniformly positive scalar curvature on noncompact 3-manifolds. In particular we show
that the Whitehead manifold lacks such a metric, and in fact that
\mathbbR3{\mathbb{R}^3} is the only contractible noncompact 3-manifold with a metric of uniformly positive scalar curvature. We also describe contractible
noncompact manifolds of higher dimension exhibiting this curvature phenomenon. Lastly we characterize all connected oriented
3-manifolds with finitely generated fundamental group allowing such a metric. 相似文献
8.
本文通过对满足Nash不等式的黎曼流形的研究,证明了对任一完备的Ricci曲率非负的n维黎曼流形,若它满足Nash不等式,且Nash常数大于最佳Nash常数,则它微分同胚于Rn. 相似文献
9.
Songting YIN 《Frontiers of Mathematics in China》2018,13(2):435-448
We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature. 相似文献
10.
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
11.
Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng. 相似文献
12.
For any complete noncompact Kähler 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
13.
Yanyan Niu 《Geometriae Dedicata》2010,149(1):363-371
In this paper, we extend the maximum principle for (1, 1)-Hermitian symmetric tensor to a complete K?hler manifold with bounded
holomorphic bisectional curvature and nonnegative orthogonal bisectional curvature. We also achieve a maximum principle for
real (p, p)-forms on a compact K?hler manifold with nonnegative holomorphic sectional curvature and vanishing Bochner tensor. 相似文献
14.
Thetheoryofharmonicmapsintoacompactnonpositivelycurvedtargetiswell-developed.TheexistenceofsuchaharmonicmappinginanygivenhomotopyclassisestablishedbyEellsandSampson[1]in1964,andlater,in1967,Hartman[2]gaveaproofoftheuniqueness.Thispaperisdevotedtoanapplicationo… 相似文献
15.
William C. Wylie 《Journal of Geometric Analysis》2006,16(3):535-550
Let (M, d) be a metric space. For 0 < r < R, let G(p, r, R) be the group obtained by considering all loops based at a point
p ∈ M whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy between them
that is contained in the open ball of radius R. In this article we study the asymptotic behavior of the G(p, r, R) groups
of complete open manifolds of nonnegative Ricci curvature. We also find relationships between the G(p, r, R) groups and tangent
cones at infinity of a metric space and show that any tangent cone at infinity of a complete open manifold of nonnegative
Ricci curvature and small linear diameter growth is its own universal cover. 相似文献
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We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing
on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the
volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume
non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum
value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the
manifold is isometric to the Euclidean space. 相似文献
18.
Yau made the following conjecture: For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional. we extend the result on the Laplace operator to that on the symmetric diffusion operator, and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finite-dimensional, when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold is nonnegative. 相似文献
19.
In this paper, we classify combinatorially different fundamental domains for any given planar discontinuous group and we give an algorithm for the complete enumeration of uniform tilings of any complete, simply connected, two-dimensional Riemannian manifold of constant curvature.Supported by Hungarian National Foundation for Scientific Research, Grant No. 1238/86. 相似文献
20.
Leonardo Biliotti Francesco Mercuri 《Bulletin of the Brazilian Mathematical Society》2014,45(3):433-452
In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature. 相似文献