共查询到20条相似文献,搜索用时 562 毫秒
1.
Keomkyo Seo 《Archiv der Mathematik》2010,94(2):173-181
We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L
2 harmonic 1-forms on M. 相似文献
2.
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
3.
Yu Ding 《Israel Journal of Mathematics》2002,129(1):241-251
In this paper we prove that when the Ricci curvature of a Riemannian manifoldM
n
is almost nonnegative, and a ballB
L
(p)⊂M
n
is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently
close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for
harmonic functions onR
n
. 相似文献
4.
Gabjin Yun 《Geometriae Dedicata》2002,89(1):133-139
Let M
n
, n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R
n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C
s
, where C
s
is the Sobolev constant for M, then there are no L
2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane. 相似文献
5.
Let M and N be two compact Riemannian manifolds. Let uk be a sequence of stationary harmonic maps from M to N with bounded energies. We may assume that it converges weakly to a weakly harmonic map u in H1,2 (M, N) as k → ∞. In this paper, we construct an example to show that the limit map u may not be stationary. © 2002 Wiley Periodicals, Inc. 相似文献
6.
Let M
m
be a compact oriented smooth manifold admitting a smooth circle action with isolated fixed points which are isolated as singularities
as well. Then all the Pontryagin numbers of M
m
are zero and its Euler number is nonnegative and even. In particular, M
m
has signature zero. We apply this to obtain non-existence of harmonic morphisms with one-dimensional fibres from various
domains, and a classification of harmonic morphisms from certain 4-manifolds.
Received: 16 May 2002 Published online: 14 February 2003
Mathematics Subject Classification (2000): 58E20, 53C43, 57R20. 相似文献
7.
Xiangao Liu 《中国科学A辑(英文版)》1999,42(11):1184-1192
The stationary for harmonic maps is considered from a Riemannian manifoldM into a complete Riemannian manifoldN without boundary, and it is proved that its singular set is contained inQ
1 2MQ
3
Project supported partially by the Development Foundation Science of Shanghai, China. 相似文献
8.
Xu Sheng Liu 《数学学报(英文版)》2010,26(2):361-368
We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0(2, 2)/SO(2) × SO(2). Let π : E →* M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type. 相似文献
9.
Gu Chao-Hao 《纯数学与应用数学通讯》1980,33(6):727-737
Let R1+1 be two-dimensional Minkowski space and M a complete Riemannian manifold of dimension n. It is proved that the solution of the Cauchy problem for the harmonic map ? : R1+1 → M exists globally. As an application to physics we conclude that the field function in a two-dimensional chiral field theory is regular for all time, if it is regular initially. 相似文献
10.
For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C
topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces. 相似文献
11.
Yawei CHU 《Frontiers of Mathematics in China》2012,7(1):19-27
Let (M
n
, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper,
by employing an elliptic estimation method, we show that (M
n
, g) is a space form if it has sufficiently small L
n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M
n
, g) with positive scalar curvature. 相似文献
12.
Joel L. Weiner 《Geometriae Dedicata》1994,53(1):25-48
Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM
2 in 4 relative to the action of the general affine group on 4. The invariants found include a normal bundle, a quadratic form onM
2 with values in the normal bundle, a symmetric connection onM
2 and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM
2, obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined. 相似文献
13.
On the singular set of stationary harmonic maps 总被引:15,自引:0,他引:15
Fabrice Bethuel 《manuscripta mathematica》1993,78(1):417-443
LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH
n−2
(Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW
1,n
(M, N), thenu is smooth. This extends results of F. Hélein for the casen=2, or the caseN is a sphere ([9], [10]). 相似文献
14.
In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian
manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative
Ricci curvature is conformally equivalent to either a plane R
2 or a cylinder R × S
1, which generalizes a theorem due to Fischer-Colbrie and Schoen [12].
The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L
2-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative
sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to
Miyaoka [20] and Palmer [21].
Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science
and Technology, Japan.
The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea.
Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga
840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea 相似文献
15.
W?odzimierz B?k 《Potential Analysis》2010,32(1):17-27
A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C
n
:n = 1,2,..}, allows for constructing measures nxC, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M
f(x)=?g ? Gf(g)nCx(g). f(x)=\sum_{\gamma\in\Gamma}f(\gamma)\nu^{\mathbf{C}}_x(\gamma). 相似文献
16.
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
17.
Armen Glebovich Sergeev 《中国科学A辑(英文版)》2008,51(4):695-706
We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. We conjecture that harmonic maps of the Riemann sphere ℂℙ1 into ΩG are related to Yang-Mills G-fields on ℝ4.
This work was partly supported by the RFBR (Grant Nos. 04-01-00236, 06-02-04012), by the program of Support of Scientific
Schools (Grant No. 1542.2003.1), and by the Scientific Program of RAS “Nonlinear Dynamics” 相似文献
18.
Wei Wang 《Annali di Matematica Pura ed Applicata》2007,186(2):359-380
By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic
Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve
the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ
n
).
Mathematics Subject Classification (2000) 53C17, 53D10, 35J70 相似文献
19.
Christian Bär 《Geometric And Functional Analysis》1996,6(6):899-942
We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS
n
,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM
1#M
2 with certain metrics is close to the union of the spectra ofM
1 and ofM
2.Partially supported by SFB 256 and by the GADGET program of the EU 相似文献
20.
Atsushi Tachikawa 《manuscripta mathematica》1992,74(1):69-81
In this paper we show a nonexistence result for harmonic maps with a rotational nondegeneracy condition from a Riemannian
manifoldM with polep
0 to a negatively curved Hadamard manifold under the condition that the metric tensor ofM is bounded and that the sectional curvature ofM at a pointp is bounded from below by −c dist(p
0,p)−2 (c: a positive constant) as dist(p
0,p)→∞.
Partly supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan 相似文献
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