共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
Włodzimierz Jelonek 《Annals of Global Analysis and Geometry》2002,22(2):119-133
The aim of this paper is to characterize Kähler surfaces in terms oftheir positive twistor bundle. We prove that an oriented four-dimensionalRiemannian manifold (M, g) admits a complex structure J compatible with the orientation and such that (M, g, J is a Kähler manifold ifand only if the positive twistor bundle (Z
+(M), g
c
) admits a verticalKilling vector field. 相似文献
3.
Renyi Ma 《manuscripta mathematica》1998,95(2):159-168
If (N, ο, J,g) is an almost K?hler manifold and M is a branched minimal immersion which is not a $J$-holomorphic curve, we show that the complex tangents are isolated and
that each has a negative index, which extends the results in the K?hler case by S. S. Chern and J. Wolfson [2] and S. Webster
[7] to almost K?hler manifolds. As an application, we get lower estimates for the genus of embedded minimal surfaces in almost
K?hler manifolds. The proofs of these results are based on the well-known Cartan's moving frame methods as in [2, 7]. In our
case, we must compute the torsion of the almost complex structures and find a useful representation of torsion. Finally, we
prove that the minimal surfaces in complex projective plane with any almost complex structure is a J-holomorphic curve if it is homologous to the complex line.
Received: 10 January 1997 / Revised version: 22 August 1997 相似文献
4.
Let (M
m
, g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove
that any bounded set of p-harmonic 1-forms in L
q
(M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology. 相似文献
5.
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
6.
Knut Smoczyk 《Mathematische Zeitschrift》2002,240(4):849-883
We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. .
Received: 11 April 2001 / Published online: 29 April 2002 相似文献
7.
Seongtag Kim 《manuscripta mathematica》2011,135(1-2):107-116
Let (M, g) be a noncompact complete n-manifold with harmonic curvature and positive Sobolev constant. Assume that the L 2 norms of the traceless Ricci curvature are finite. We prove that (M, g) is Einstein if n ?? 5 and the L n/2 norms of the Weyl curvature and traceless Ricci curvature are small enough. 相似文献
8.
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
9.
In this paper, we prove that if M is a K?hler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and
the curvature is positive somewhere, then the K?hler-Ricci flow converges to a K?hler-Einstein metric with constant bisectional
curvature. In a subsequent paper [7], we prove the same result for general K?hler-Einstein manifolds in all dimension. This
gives an affirmative answer to a long standing problem in K?hler Ricci flow: On a compact K?hler-Einstein manifold, does the
K?hler-Ricci flow converge to a K?hler-Einstein metric if the initial metric has a positive bisectional curvature? Our main
method is to find a set of new functionals which are essentially decreasing under the K?hler Ricci flow while they have uniform
lower bounds. This property gives the crucial estimate we need to tackle this problem.
Oblatum 8-IX-2000 & 30-VII-2001?Published online: 19 November 2001 相似文献
10.
Gang Liu 《Geometric And Functional Analysis》2014,24(5):1591-1607
Let (M n , g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N k with c 1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds. 相似文献
11.
A construction due to Sym and Bobenko recovers constant mean curvature surfaces in euclidean 3-space from their harmonic Gauss maps. We generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and the sphere (the target space of the Gauss map) by a Kähler symmetric space of compact type with its standard embedding into the Lie algebra ${\mathfrak{g}}A construction due to Sym and Bobenko recovers constant mean curvature surfaces in euclidean 3-space from their harmonic Gauss
maps. We generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and
the sphere (the target space of the Gauss map) by a K?hler symmetric space of compact type with its standard embedding into
the Lie algebra
\mathfrakg{\mathfrak{g}} of its transvection group. Thus we obtain a new class of immersed K?hler submanifolds of
\mathfrakg{\mathfrak{g}} and we derive their properties. 相似文献
12.
Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar
curvature. Let r = 1 if n = 2,3 and r = 2[n/2]-1 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small
Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.
相似文献
13.
E. Calvi?o-Louzao E. García-Río M. E. Vázquez-Abal R. Vázquez-Lorenzo 《Annals of Global Analysis and Geometry》2012,41(2):241-252
It is shown that indefinite strictly almost K?hler and opposite K?hler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drăghici from the positive
definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit
strictly almost paraK?hler and opposite paraK?hler structures
(\mathfrakJ,\mathfrakJ¢){(\mathfrak{J},\mathfrak{J}^{\prime})} with
\mathfrakJ{\mathfrak{J}} -invariant Ricci operator are shown. 相似文献
14.
Craig van Coevering 《Mathematische Annalen》2010,347(3):581-611
We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in ${H^2_c(Y,\mathbb{R})}We prove that a crepant resolution π : Y → X of a Ricci-flat K?hler cone X admits a complete Ricci-flat K?hler metric asymptotic to the cone metric in every K?hler class in
H2c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A K?hler cone (X,[`(g)]){(X,\bar{g})} is a metric cone over a Sasaki manifold (S, g), i.e. ${X=C(S):=S\times\mathbb{R}_{ >0 }}${X=C(S):=S\times\mathbb{R}_{ >0 }} with [`(g)]=dr2 +r2 g{\bar{g}=dr^2 +r^2 g}, and (X,[`(g)]){(X,\bar{g})} is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat K?hler metrics on
crepant resolutions
p:Y? X=\mathbbCn /G{\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}, with
G ì SL(n,\mathbbC){\Gamma\subset SL(n,\mathbb{C})}, due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric K?hler cone admits a Ricci-flat K?hler
cone metric. It follows that if a toric K?hler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T
n
-invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in
H2c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry. 相似文献
15.
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
16.
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
17.
In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schr?dinger
flow for maps from a compact Riemannian manifold M with dim(M) ≤ 3 into a compact K?hler manifold (N, J) with nonpositive Riemannian sectional curvature
Received November 1, 1999, Revised January 14, 2000, Accepted March 29, 2000 相似文献
18.
Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold 总被引:1,自引:0,他引:1
Marian Ioan Munteanu 《Mediterranean Journal of Mathematics》2008,5(1):43-59
In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure
of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find
an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from
T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient
structures on the unit tangent bundle and the tangent sphere bundle, respectively.
This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania. 相似文献
19.
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
20.
Yuguang ZHANG 《数学年刊B辑(英文版)》2007,28(4):421-428
Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where X j is a Calabi-Yau manifold, or a hyperKähler manifold, or X j satisfies H 0(X j , Ω p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J ∈, g ∈) on M such that the volume \({\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V\), the sectional curvature |K(g ∈)| < Λ2, and the Ricci-tensor Ric(g ∈)> ?∈g ∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where X i is a Calabi-Yau manifold, or a hyperKähler manifold, or X i satisfies H 0(X i , Ω p ) = {0}, p > 0. 相似文献