共查询到20条相似文献,搜索用时 15 毫秒
1.
Existence of solution for semilinear problem with the Laplace-Beltrami operator on non-compact Riemannian manifolds with rich
symmetries is proved by concentration compactness based on actions of the manifold's isometry group. 相似文献
2.
For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As
an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The
main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the
volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which
density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial
growth of the volume function. 相似文献
3.
Michael Taylor 《Journal of Geometric Analysis》2007,17(2):365-374
We establish further regularity of the Cα and H1,p limits of smooth, n-dimensional Riemannian manifolds with a lower bound on Ricci tensor and injectivity radius, and an upper
bound on volume, first considered in [1]. We use this extra regularity to show that such a limit is a nonbranching geodesic
space, as defined in [10], and to construct a variant of a geodesic flow for such a limit. We contrast the behavior of some
slightly more singular limits. 相似文献
4.
The conformal class of a Hermitian metric g on a compact almost complex manifold (M2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted
version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature sJ. We ask if the conformal class of g carries a metric with constant sJ, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the
affirmative. In fact, we show that (2m−1)sJ−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution
of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant
is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal
bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that
when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian
Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere
equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the
conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well.
We discuss some applications. 相似文献
5.
Christine M. Guenther 《Journal of Geometric Analysis》2002,12(3):425-436
In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u,
on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent
Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold.
We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature
bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution
of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel. 相似文献
6.
We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n=2p, p≥2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q∈p, 0≤q≤n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M′ of dimension n≥6, which are isospectral on functions and such that βp(M)<βp(M’), for 0
4 and ? 2 2 , respectively. 相似文献
7.
In this article, we study closed Riemannian manifolds with small excess. We show that a closed connected Riemannian manifold
with Ricci curvature and injectivity radius bounded from below is homeomorphic to a sphere if it has sufficiently small excess.
We also show that a closed connected Riemannian manifold with weakly bounded geometry is a homotopy sphere if its excess is
small enough. 相似文献
8.
We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan).
In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved.
Supported by a JSPS postdoctoral fellowship. 相似文献
9.
In this paper we shall prove some results pertaining to the existence and multiplicity of normal geodesics joining two given
submanifolds of an orthogonal splitting Lorentzian manifold. To this aim, we look for critical points of an unbounded suitable
functional by using a Saddle-Point Theorem and the relative category theory. 相似文献
10.
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
11.
Ruth Gornet 《Journal of Geometric Analysis》2000,10(2):281-298
The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions
but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length
spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse,
as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral.
The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian
nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral
deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method,
a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and
Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method,
and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson
method.
To the memory of Hubert Pesce, a valued friend and colleague. 相似文献
12.
Seok-Ku Ko 《Journal of Geometric Analysis》1999,9(1):119-141
For the Riemann surface of the topological type, we can get a conformai model in orientable Riemannian manifolds. We will
prove that there is a conformally equivalent model in orientable Riemannian manifolds for a given open Riemann surface. To
end up we utilize Garsia 's Continuity lemma and Brouwer's Fixed Point lemma along with the Teichmüller theory. 相似文献
13.
In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic
in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound
on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower
bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also
give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first
conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum
and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families
of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known
methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum.
In memory of Robert Brooks 相似文献
14.
Ye Li 《Journal of Geometric Analysis》2007,17(3):495-511
We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor
having finite L2 norm in dimension 4. 相似文献
15.
In the context of a complete simply connected Riemannian manifold of pinched negative curvature, we show that several families
of approach domains are equivalent for convergence to points of the boundary, and for the purposes of Hp-theory. 相似文献
16.
We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra
that models fibered cusps at infinity. Using tracelike functionals that generate the 0-dimensional Hochschild cohomology groups
we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior
of X and a term that comes from the boundary. This leads to an abstract answer to the index problem formulated in [11]. We
give a more precise answer for firstorder differential operators when the base of the boundary fiber bundle is S1. In particular, for Dirac operators associated to a metric of the form
near ∂X = {x = 0} with twisting bundle T we obtain
in terms of the integral of the Atiyah-Singer form in the interior of X, and the adiabatic limit of the η-invariant of the
restriction of the operator to the boundary. 相似文献
17.
We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation
contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov
showed that there are no nontrivial isospectral deformations of such metrics. 相似文献
18.
We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic
is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show
that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us
to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow. 相似文献
19.
Ursula Hamenstädt 《Journal of Geometric Analysis》2004,14(2):281-290
Let M be a complete geometrically finite manifold of bounded negative curvature, infinite volume, and dimension at least 3.We give both a lower bound for the bottom of the spectrum of M and an upper bound for the number of the small eigenvalues
of M. These bounds only depend on the dimension, curvature bounds and the volume of the oneneighborhood of the convex core. 相似文献
20.
We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we
produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different
weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples
of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These
results follow from a method that uses integral roots of the Krawtchouk polynomials.
We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence
we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does
not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds.
We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with
different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed
geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0. 相似文献