共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
Pierre Albin 《Advances in Mathematics》2007,213(1):1-52
We study the Gauss-Bonnet theorem as a renormalized index theorem for edge metrics. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence and the asymptotically cylindrical metrics of the Atiyah-Patodi-Singer index theorem. We use renormalization to make sense of the curvature integral and the dimensions of the L2-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod xm, we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function. 相似文献
3.
Neil S. Trudinger Xu-Jia Wang 《Calculus of Variations and Partial Differential Equations》2009,35(3):317-338
In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g
0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g
0) is not conformally equivalent to the unit sphere S
n
and that the set of all such metrics is compact. When (M, g
0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence
theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the
solvability of the k-Yamabe problem for k > n/2.
This work was supported by the Australian Research Council. 相似文献
4.
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
5.
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
6.
In this article we study the first eigenvalue of the Laplacian on a compact manifold using stable bundles and balanced bases. Our main result is the following: Let M be a compact Kähler manifold of complex dimension n and E a holomorphic vector bundle of rank r over M. If E is globally generated and its Gieseker point Te is stable, then for any Kähler metric g on M\(\lambda _1 (M,g) \leqslant \frac{{4\pi h^0 (E)}}{{r(h^0 (E) - r)}} \cdot \frac{{\left\langle {C_1 (E) \cup [\omega ]^{n - 1} ,[M]} \right\rangle }}{{(n - 1)!vol(M,[\omega ])}}\) where ω = ωg is the Kähler form associated to g.By this method we obtain, for example, a sharp upper bound for λ1 of Kähler metrics on complex Grassmannians. 相似文献
7.
Carolyn S. Gordon 《Inventiones Mathematicae》2001,145(2):317-331
We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R
n
for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be
positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close
to the flat metric.
Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001 相似文献
8.
Cs. Vincze 《Differential Geometry and its Applications》2006,24(1):1-20
As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F−1(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian. 相似文献
9.
András Vasy 《Advances in Mathematics》2010,223(1):49-97
In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y−. 相似文献
10.
Xu-Qian Fan 《manuscripta mathematica》2006,120(4):435-467
One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)2. We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality. 相似文献
11.
We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to
the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3).
The choice of a splitting
\mathfrakg* = V1 ?V2 {\mathfrak{g}^*} = {V_1} \oplus {V_2} , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions
for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction
to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that
admit a hypo structure. 相似文献
12.
Pierre Jammes 《manuscripta mathematica》2007,123(1):15-23
Let M
n
be an n-dimensional compact manifold, with n ≥ 3. For any conformal class C of riemannian metrics on M, we set , where μ
p,k
(M,g) is the kth eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that . We also prove that if g is a smooth metric such that , and n = 0,2,3 mod 4, then there is a non-zero corresponding eigenform of degree with constant length. As a corollary, on a four-manifold with non vanishing Euler characteristic, there is no such smooth
extremal metric. 相似文献
13.
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
14.
15.
We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K?hler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in ${SL(n,\mathbb{H})}$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K?hler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K?hler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics. 相似文献
16.
We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:
- (a)
- The metric and the external perturbation are smooth enough.
- (b)
- The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.
- (c)
- The frequency of the external perturbation is Diophantine.
- (d)
- The external potential satisfies a generic condition depending on the periodic orbit considered in (b).
17.
Brian Clarke Yanir A. Rubinstein 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric. 相似文献
18.
Vladimir Sharafutdinov 《Journal of Geometric Analysis》2007,17(1):147-187
The Dirichlet-to-Neumann (DN) map Λg: C∞ (?M) → C∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λgh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If gt = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λg t = Λg + tΛ? + o(t). We study the question: How do tensor fields ? look like for which Λ? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold. 相似文献
19.
Yoshinobu Kamishima 《Annals of Global Analysis and Geometry》1994,12(1):261-289
A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday 相似文献
20.
In this paper we prove that given a smoothly conformally compact asymptotically hyperbolic metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact and asymptotically hyperbolic. We adapt recent results of Schnürer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric. 相似文献