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1.
We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface
M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space,
we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L
2 harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant.
Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional
curvature to be stable. 相似文献
2.
Qiaoling Wang 《Annals of Global Analysis and Geometry》2010,37(2):113-124
We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has
only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem
for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a
result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end
to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the
same kind of submanifolds are also obtained. 相似文献
3.
We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric
or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R
2 has non-negative isotropic curvature.
The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author
was partially supported by NSF grant DMS-0604960. 相似文献
4.
Let M n be a complete oriented noncompact hypersurface in a complete Riemannian manifold N n+1 of nonnegative sectional curvature with ${2 \leq n \leq 5}$ . We prove that if M satisfies a stability condition, then there are no non-trivial L 2 harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of ${\mathbb{R}}$ and a compact manifold with nonnegative sectional curvature. In case ${n \geq 5}$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface. 相似文献
5.
Changyu Xia 《manuscripta mathematica》1994,85(1):79-87
LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity
theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to
the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound.
Supported by the JSPS postdoctoral fellowship and NSF of China 相似文献
6.
Matheus Vieira 《Archiv der Mathematik》2013,101(6):581-590
In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Émery–Ricci curvature if the space of weighted L 2 harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface. 相似文献
7.
Peng Zhu 《Annals of Global Analysis and Geometry》2011,40(4):427-434
We prove that L
2 harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L
2 harmonic two-form. We obtain that an almost K?hler manifold of non-negative isotropic curvature is K?hler and a symplectic
manifold can not admit any almost K?hler structure of positive isotropic curvature. 相似文献
8.
Huiling Le 《Probability Theory and Related Fields》1999,114(1):85-96
Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c
1/{r
2 ln r} and below by −c
2
r
2, where c
1 and c
2 are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a
κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire
sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of
κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M.
Received: 19 September 1997 相似文献
9.
Bin XU 《数学年刊B辑(英文版)》2007,28(2):195-204
We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology. 相似文献
10.
Yi Bing SHEN Xiao Hua ZHU 《数学学报(英文版)》2005,21(3):631-642
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1+1 are considered. 相似文献
11.
Eigenvalue Monotonicity for the Ricci-Hamilton Flow 总被引:4,自引:0,他引:4
Li Ma 《Annals of Global Analysis and Geometry》2006,29(3):287-292
In this short note, we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow.
Mathematics Subject Classifications (1991): 53C44
In Memory of S.S. Chern 相似文献
12.
I. G. Nikolaev 《Commentarii Mathematici Helvetici》1995,70(1):210-234
Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural
to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant
curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of
this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL
1-small integral anisotropy haveL
p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that
of constant curvature in theW
p
2
-norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability
results are based on the generalization of Schur' theorem to metric spaces. 相似文献
13.
Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion. 相似文献
14.
Tamás Rapcsák 《Journal of Global Optimization》2008,40(1-3):375-388
The aim of the paper is to show how to explicitly express the function of sectional curvature with the first and second derivatives
of the problem’s functions in the case of submanifolds determined by equality constraints in the n-dimensional Euclidean space endowed with the induced Riemannian metric, which is followed by the formulation of the minimization
problem of sectional curvature at an arbitrary point of the given submanifold as a global minimization one on a Stiefel manifold.
Based on the results, the sectional curvatures of Stiefel manifolds are analysed and the maximal and minimal sectional curvatures
on an ellipsoid are determined.
This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA-T043276 and OTKA-K60480. 相似文献
15.
Jui-Tang Ray Chen 《Annals of Global Analysis and Geometry》2009,36(2):161-190
This article concerns the structure of complete noncompact stable hypersurfaces M
n
with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N
n+1. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M
n
, n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with
, respectively, has only one end. 相似文献
16.
Huiling Le 《Probability Theory and Related Fields》1996,106(1):137-149
Summary. Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse
of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of
the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire
S
m−1
. This improves a result of Hsu and March.
Received: 7 December 1994/In revised form: 2 September 1995 相似文献
17.
This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.Mathematics Subject Classiffications (2000). 17A70, 53C29, 53C99, 57Rxx 相似文献
18.
Qilin Yang 《Differential Geometry and its Applications》2007,25(1):1-7
It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal. 相似文献
19.
Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is
an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary
isometric to a round sphere, is necessarily a flat disc.
Received: 2 February 2002; in final form: 1 August 2002 /
Published online: 1 April 2003
Mathematics Subject Classification (1991): 53C27, 53C40, 53C80, 58G25
The authors would like to thank Lars Andersson for helpful discussions and for bringing to our knowledge the information
regarding Remark 4. We are also grateful to the referee for pointing out that Corollary 5 and Corollary 6 are only valid when
the boundary is at least 2-dimensional.
Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967 相似文献
20.
We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian
manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the
volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a2 and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)2/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only.
Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY
Research Award Program. 相似文献