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1.
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
2.
We show that a noncompact, complete, simply connected harmonic manifold (M d, g) with volume densityθ m(r)=sinhd-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M 2d, g) with volume densityθ m(r)=sinh2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented. 相似文献
3.
Georgios Daskalopoulos Chikako Mese Alina Vdovina 《Geometric And Functional Analysis》2011,21(4):905-919
In this paper, we study the behavior of harmonic maps into complexes with branching differentiable manifold structure. The
main examples of such target spaces are Euclidean and hyperbolic buildings. We show that a harmonic map from an irreducible
symmetric space of noncompact type other than real or complex hyperbolic into these complexes are non-branching. As an application,
we prove rank-one and higher-rank superrigidity for the isometry groups of a class of complexes which includes hyperbolic
buildings as a special case. 相似文献
4.
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
5.
Bo-Yong Chen 《Arkiv f?r Matematik》2013,51(2):269-291
We give first of all a new criterion for Bergman completeness in terms of the pluricomplex Green function. Among several applications, we prove in particular that every Stein subvariety in a complex manifold admits a Bergman complete Stein neighborhood basis, which improves a theorem of Siu. Secondly, we give for hyperbolic Riemann surfaces a sufficient condition for when the Bergman and Poincaré metrics are quasi-isometric. A consequence is an equivalent characterization of uniformly perfect planar domains in terms of growth rates of the Bergman kernel and metric. Finally, we provide a noncompact Bergman complete pseudoconvex manifold without nonconstant negative plurisubharmonic functions. 相似文献
6.
Jaap Eldering 《Comptes Rendus Mathematique》2012,350(11-12):617-620
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof. 相似文献
7.
HARMONIC FUNCTIONS ON A COMPLETE NONCOMPACT MANIFOLD WITH ASYMPTOTICALLY NONNEGATIVE CURVATURE 总被引:4,自引:0,他引:4 下载免费PDF全文
The authors prove the space of harmonic functions with polynomial growth of a fixed rate on a complete noncompact Riemannian manifold with asymptotically nonnegative curvature is finite dimensional. 相似文献
8.
Shoshichi Kobayashi 《Results in Mathematics》2001,40(1-4):246-256
One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (in the sense that its intrinsic pseudo-distance is a (complete) distance) is that it admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. The concept of hyperbolicity can be readily extended to almost complex manifolds. We will show that the above result for hyperbolicity can be generalized to the almost complex case. As an application, we prove that every point of an almost complex manifold has a complete hyperbolic neighborhood. In real dimension 4, this fact was established by Debalme and Ivashkovich [2] by a completely different method. 相似文献
9.
In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature
is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on
the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound,
we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover,
by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with
large volume growth. 相似文献
10.
We prove that, on a complete noncompact Riemannian manifold with bounded geometry, the Lp boundedness of the Riesz transform, for p>2, is stable under a quasi-isometric and integrable change of metric. As an intermediate
step, we treat the case of weighted divergence form operators in the Euclidean space. 相似文献
11.
This paper consists of two results dealing with balanced metrics (in Donaldson terminology) on noncompact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan–Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in Loi and Zedda (Mathematische Annalen, 2011, to appear) we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that α g is not balanced for all α > 0. 相似文献
12.
Wan-Xiong Shi 《Journal of Geometric Analysis》1998,8(1):117-142
In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ?n. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22]. 相似文献
13.
Satyaki Dutta 《Advances in Mathematics》2010,224(2):525-538
In this paper, we prove that under a lower bound on the Ricci curvature and an assumption on the asymptotic behavior of the scalar curvature, a complete conformally compact manifold whose conformal boundary is the round sphere has to be the hyperbolic space. It generalizes similar previous results where stronger conditions on the Ricci curvature or restrictions on dimension are imposed. 相似文献
14.
Matthew B. Stenzel 《Journal of Fourier Analysis and Applications》2009,15(6):839-856
Let f be a rapidly decreasing radial function on a Riemannian symmetric space of noncompact type whose spherical Fourier transform
has compact support. We prove a reconstruction theorem which recovers f from the values of an integral operator applied to f on a discrete subset. When G/K is of the complex type we prove a sampling formula recovering f from its own values on a discrete subset. We give explicit results for three dimensional hyperbolic space. 相似文献
15.
Qi S. Zhang 《Journal of Functional Analysis》2012,263(7):2051-2101
Let M be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (1.1) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This result has the following consequences. 1. It seems to give the first example of connected, complete manifolds with bounded geometry where a standard Log Sobolev inequality does not have an extremal. 2. It gives a negative answer to the open question on the existence of extremal of Perelman?s W entropy in the noncompact case, which was stipulated by Perelman (2002) [22, p. 9, 3.2 Remark]. 3. It helps to prove, in some cases, that noncompact shrinking breathers of Ricci flow are gradient shrinking solitons. 相似文献
16.
Jiaping Wang 《Journal of Geometric Analysis》1998,8(3):485-514
We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between
two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application,
we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary
maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing
energy density. 相似文献
17.
We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive
sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic
or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions.
In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the
Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic.
Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one
containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification
of finite volume quotients of the complex hyperbolic space.
Oblatum 2-IX-1994 & 7-VIII-1995 相似文献
18.
We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily-Borel resp. Siu-Yau compactification of finite volume quotients of the complex hyperbolic space.Oblatum 2-IX-1994 & 7-VIII-1995 相似文献
19.
Bogdan Alexandrov 《Annals of Global Analysis and Geometry》2002,22(1):75-98
We call a quaternionic Kähler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kähler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kähler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kähler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space. 相似文献
20.
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