Positive Eigenfunctions of the Laplacian and Conformal Densities on Homogeneous Trees (original) (raw)

The distribution of radial eigenvalues of the Euclidean Laplacian on homogeneous isotropic trees

Complex Analysis and its Synergies, 2021

Let Ω be a bounded open subset of R n and ∆ the Euclidean Laplace operator n i=1 ∂ 2 /∂x 2 i. Let β(x) denote the number of eigenvalues less or equal to x with respect to the eigenvalue problem ∆f = −xf on Ω with f = 0 on the boundary of Ω. A well-known result due to Hermann Weyl gives the asymptotic formula β(x) = (2π) −n Bnmn(Ω)x n/2 + o(x n/2) as x → ∞, where Bn is the volume of the unit ball in R n and mn(Ω) is the volume of Ω. In this work, we consider the analogous problem for radial functions in the discrete setting of the homogeneous isotropic tree T of homogeneity q + 1 (q ≥ 2). As the volume of T with respect to the hyperbolic metric is infinite, we don't expect and indeed we show that there is no analogous result for the commonly-used hyperbolic Laplacian on T. We consider instead the eigenvalue problem for radial functions on T with respect to the Euclidean Laplacian on T introduced in [6], where the boundary condition f = 0 means that f converges radially to 0 at ∞. We prove that β(x) is within 2 of log q √ x. We also consider other boundary conditions and pose some open questions.

SPHERICAL FUNCTIONS AND CONFORMAL DENSITIES ON SPHERICALLY SYMMETRIC CAT( 1)-SPACES

1995

| Let X be a CAT(?1)? space which is spherically symmetric around some point x 0 2 X and whose boundary has nite positive s?dimensional Hausdor measure. Let = ( x ) x2X be a conformal density of dimension d > s=2 on @X. We prove that x 0 is a weak limit of measures supported on spheres centered at x 0 . These measures are expressed in terms of the total mass function of and of the d?dimensional spherical function on X. In particular, this result proves that is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.

Measures of Dirichlet type on regular polygons and their moments

Journal of Approximation Theory, 1992

The (k-I)-dimensional simplex is projected onto the convex hull of the kth roots of unity in C, and a dihedral-group-invariant Dirichlet-type measure is thereby constructed. The integrals of monomials z'"?" are obtained as single sums. A certain radial measure on the disc is obtained as a weak-*limit.

Determinants of Laplacians and isopolar metrics on surfaces of infinite area

Duke Mathematical Journal, 2003

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances. Contents 1. Introduction 2. Definition of the determinant 3. Properties of the determinant in the hyperbolic case 4. Relative determinant 4.1. Conformal deformation theory 4.2. Zeta-regularization of the relative determinant 4.3. Polyakov formula 5. Hadamard factorization of the relative determinant 5.1. Divisor of the relative determinant 5.2. Growth estimates on the relative determinant 5.3. Asymptotics and order 6. Compactness for isopolar classes Appendix A. Finiteness and properness for hyperbolic surfaces Appendix B. Resolvent construction and estimates Appendix C. Logarithmic derivative of the relative scattering operator Appendix D. Isopolar Surfaces, by Robert Brooks References

On conformal divergences and their population minimizers

Total Bregman divergences are recent regularizations of Bregman divergences that are invariant to rotations of the natural space, a very desirable property in many signal processing applications. Clustering with total Bregman divergences often display superior results compared to ordinary Bregman divergences (Liu, 2011), and raises, for a wide applicability in clustering, the question of the complete characterization of the left and right population minimizers -total Bregman divergences are indeed not symmetric.

Estimates for Functions of the Laplace Operator on Homogeneous Trees

In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest{neighbour) Laplacian. We nd pointwise esti- mates for the heat and resolvent kernels, and the Lp-Lq mapping properties of the corresponding operators. M. M. H. Pang (P) considered the semigroup (e t)t>0: In this paper, we consider the heat semigroup associated to the canonical Laplace operator on a homogeneous tree, which is perhaps the basic example of a graph where the cardinality of the \ball of radiusr" grows exponentially in r. We study the properties of the heat and resolvent operators, and discover very close analogies with diusions on hyperbolic spaces, in line with the philosophy of the above-mentioned authors. A homogeneous tree X of degree q + 1 is dened to be a connected graph with no loops, in which every vertex is adjacent to q + 1 other vertices. We denote by d the natural distance on X ;d (x;y) being the number of edges between the vertices x and y; and by Lp(X)...

Jones-Makarov's" Density of harmonic measure" revisited

2006

We study the boundary properties of the Green function of bounded simply connected domains in the plane. Essentially, this amounts to studying the conformal mapping taking the unit disk onto the domain in question. Our technique is inspired by a 1995 paper of Jones and Makarov. The main tools are an integral identity as well as a uniform Sobolev imbedding theorem. The latter is in a sense dual to the exponential integrability of Marcinkiewicz-Zygmund integrals. We also develop a Grunsky identity, which contains the information of the classical Grunsky inequality. This Grunsky identity is the case p = 2 of a more general Grunsky identity for L p spaces.

Heat kernel expansions, ambient metrics and conformal invariants

Advances in Mathematics

The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family H(r; g) of self-adjoint elliptic differential operators. H(r; g) is a non-Laplace-type perturbation of the conformal Laplacian P 2 (g) = H(0; g). It is defined in terms of the metric g and covariant derivatives of the curvature of g. We study the heat kernel coefficients a 2k (r; g) of H(r; g) on closed manifolds. We prove general structural results for the heat kernel coefficients a 2k (r; g) and derive explicit formulas for a 0 (r) and a 2 (r) in terms of renormalized volume coefficients. The Taylor coefficients of a 2k (r; g) (as functions of r) interpolate between the renormalized volume coefficients of a metric g (k = 0) and the heat kernel coefficients of the conformal Laplacian of g (r = 0). Although H(r; g) is not conformally covariant, there is a beautiful formula for the conformal variation of the trace of its heat kernel. Its proof rests on the combinatorial relations between Taylor coefficients of H(r; g) and GJMS-operators. As a consequence, we give a heat equation proof of the conformal transformation law of the integrated renormalized volume coefficients. By refining these arguments, we also give a heat equation proof of the conformal transformation law of the renormalized volume coefficients itself. The Taylor coefficients of a 2 (r) define a sequence of higher-order Riemannian curvature functionals with extremal properties at Einstein metrics which are analogous to those of integrated renormalized volume coefficients. Among the various additional results the reader finds a Polyakov-type formula for the renormalized volume of a Poincaré-Einstein metric in terms of Q-curvature of its conformal infinity and additional holographic terms. The present study of relations between spectral theoretic heat kernel coefficients and geometric quantities like renormalized volume coefficients is stimulated by the holographic perspective of the AdS/CFT-duality (proposing relations between functional determinants and renormalized volumes). ANDREAS JUHL 7. The Gaussian integral 34 8. A conformal primitive of the Gaussian integral 41 9. Proof of Theorem 1.4 44 10. Proof of Theorem 1.8 44 11. Proof of Theorem 1.7 49 12. Extremal properties of some curvature functionals 52 13. Further results, comments and open problems 60 13.1. Global conformal invariants 60 13.2. Spectral zeta functions 61 13.3. A heat kernel proof of the variational formula of v(r) 65 13.4. Hessians 67 13.5. Spectral geometry 70 13.6. Relation to Q-curvature 71 13.7. First-order perturbation 72 13.8. Quantization and Einstein condition 72 13.9. Non-Laplace-type operators 72 14. Appendix 72 14.1. The holographic Laplacian 73 14.2. The heat kernel coefficients a 0 , a 2 and a 4 73 14.3. The heat kernel coefficient a 6 74 14.4. a 6 for locally conformally flat metrics 79 14.5. Holographic formulas for heat kernel coefficients and applications 82 14.6. Polyakov formulas for functional determinants of P 2 89 14.7. Polyakov type formulas for renormalized volumes 90 14.8. Heat kernel coefficients of round spheres and hyperbolic spaces 93 14.9. The correction terms as polynomials in v 2k 95 14.10. Product metrics and consequences 95 14.11. On the coefficient a 4 (r) 99 References 107

On polygonal measures with vanishing harmonic moments

Journal d'Analyse Mathématique, 2014

A polygonal measure is the sum of finitely many real constant density measures supported on triangles in C. Given a finite set S ⊂ C, we study the existence of polygonal measures spanned by triangles with vertices in S, which have all harmonic moments vanishing. For S generic, we show that the dimension of the linear space of such measures is |S|−3 2 .