Space Hn Research Articles

Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature

The paper considers stationary critical points of the heat flow in sphere SN and in hyperbolic space HN, and proves several results corresponding to those in Euclidean space RN which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x0, is fixed and initial-boundary value problems are considered for the heat equation on bounded domains containing x0. It is shown that for any initial data satisfying the balance law with respect to x0 (or being centrosymmetric with respect to x0) the corresponding solution always has x0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x0 (or is centrosymmetric with respect to x0, respectively).

On the global regularity of wave maps in the critical Sobolev norm

We extend the recent result of T. Tao [6] to wave maps defined from the Minkowski space Rn+1, n ≥ 5, to a target manifold N which possesses a “bounded parallelizable” structure. This is the case of Lie groups, homogeneous spaces as well as the hyperbolic spaces HN. General compact Riemannian manifolds can be imbedded as totally geodesic submanifolds in bounded parallelizable manifolds, see [1], and therefore are also covered, in principle, by our result. Compactness of the target manifold, which seemed to play an important role in [6], turns out however to play no role in our discussion. Our proof follows closely that of [6] and is based, in particular, on its remarkable microlocal gauge renormalization idea.

The Power-Compositions Determinant and Its Application to Global Optimization

Let C(n,p) be the set of p-compositions of an integer n, i.e., the set of p-tuples $\alpha=(\alpha_1,\ldots,\alpha_p)$ of nonnegative integers such that $\alpha_1+\cdots+\alpha_p=n$. The main result of this paper is an explicit formula for the determinant of the matrix whose entries are $\alpha^{\beta}=\alpha_1^{\beta_1}\cdots\alpha_p^{\beta_p}$ where $\alpha,\beta\in C(n,p)$. The formula shows that the determinant is positive and has a nice factorization. As an application, it is shown that the polynomials $p_\alpha(x)=(\alpha_1x_1+\cdots+\alpha_px_p)^n$ with $\alpha\in C(n,p)$ form a basis of the vector space Hn[x1 , . . . ,xp] of homogeneous polynomials of degree n in p variables. The result is of interest in the context of global optimization because it allows an explicit representation of polynomials as a difference of convex functions.

COHOMOLOGY RINGS OF TORIC HYPERKÄHLER MANIFOLDS

A hyperKähler quotient of a quaternionic vector space HN by a subtorus of TN is called a toric hyperKähler manifold if it has a manifold structure. We describe the cohomology ring of a toric hyperKähler manifold in two ways. Since its topology depends only on the subtorus, we describe its cohomology ring only in terms of the subtorus. On the other hand, a toric hyperKähler manifold is constructed from a certain arrangement of hyperplanes. So we also describe its cohomology ring in terms of the arrangement of hyperplanes.

Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N−1)-dimensional Lobachevsky space HN−1, N=n+l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N−2)-dimensional sphere SN−2 by pointlike sources. Some examples with billiards of finite volume and hence oscillating behavior near the singularity are considered. Among them examples with square and triangle two-dimensional billiards (e.g., that of the Bianchi-IX model) and a four-dimensional billiard in “truncated” D=11 supergravity model (without the Chern–Simons term) are considered. It is shown that the inclusion of the Chern–Simons term destroys the confining of a billiard.

Harmonic Analysis for Spinor Fields in Complex Hyperbolic Spaces

L2 harmonic analysis for Dirac spinors on the complex hyperbolic space Hn(C) is developed. The spinor spherical functions are calculated in terms of Jacobi functions. The Plancherel and Paley–Wiener theorems for the spherical transform are obtained by reduction to Jacobi analysis. We demonstrate analytically the existence of harmonic L2 spinors in the case of n even. The action of the invariant differential operators on the Poisson transforms is given explicitly.

Logarithmic Derivatives of Heat Kernels and Logarithmic Sobolev Inequalities with Unbounded Diffusion Coefficients on Loop Spaces

In this paper, we will give a sufficient condition on the logarithmic derivative of the heat kernel under which a logarithmic Sobolev inequality (LSI, in abbreviation) on a loop space holds. As an application, we prove an LSI on a pinned path space over the hyperbolic space Hn with constant sectional curvature −a (a⩾0). The diffusion coefficient of the Dirichlet form is an unbounded but exponentially integrable function. Applying to the case when a=0, we can prove an LSI with a logarithmic Sobolev constant 18 in the case of standard pinned Brownian motion. Using the LSI on the pinned path space on Hn, we will prove an LSI on each homotopy class of the loop space over a constant negative curvature compact Riemannian manifold.

Quadratic Integers and Coxeter Groups

AbstractMatrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive n-space or hyperbolic (n+1)-space Hn+1. For small n, thesemay be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of Hn+1. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group PSL2(), the Gaussian modular (Picard) group PSL2([i]), and the Eisenstein modular group PSL2([ω]).

Rooks on ferrers boards and matrix integrals

Let $$C(n, N) = \smallint _{H_N } tr Z^{2n} \mu (dZ)$$ denote a matrix integral over a U(N)- invariant Gaussian measure Μ on the space Hn of Hermitian N×N matrices. The integral is known to be always a positive integer. We derive a simple combinatorial interpretation of this integral in terms of rook configurations on Ferrers boards. The formula $$C(n,N) = (2n - 1)!! \sum\limits_{k = 0}^n {\left( {\mathop {k + 1}\limits^N } \right) \left( {\mathop k\limits^n } \right) } 2^k ,$$ found by J. Harer and D. Zagier, immediately follows from our interpretation.

A construction of Killing spinors on Sn

We derive simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n. Using these results we also construct the Killing spinors on various AdS×Sphere supergravity backgrounds, including AdS5×S5, AdS4×S7, and AdS7×S4. In addition, we extend previous results to obtain the Killing spinors on the hyperbolic spaces Hn.

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space Hn, n > 2. For ν > 0, the Brownian bridge B(ν) of length ν on H is the process Bt, 0 ≤t≤ν, conditioned by B0 = Bν = o, where o is an origin in H. It is proved that the process \(\) converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ3). The same result holds for the simple random walk on an homogeneous tree.

A Bernstein‐type theorem in hyperbolic spaces

The following Bernstein-type theorem in hyperbolic spaces is proved. Let ∑ be a non-zero constant mean curvature complete hypersurface in the hyperbolic space ℍn. Suppose that there exists a one-to-one orthogonal projection from ∑ into a horosphere. (1) If the projection is surjective, then ∑ is a horosphere. (2) If the projection is not surjective and its image is simply connected, then ∑ is a hypersphere.

On the Area Growth of Minimal Surface in Hn

We prove that a properly immersed complete minimal surface in hyperbolic space Hn has minimal area growth if some weighted L2-norm of its second fundamental form is finite. The converse is also proved, provided the surface has finite topological type. An analogue of Osserman′s inequality is also obtained.

The Heat Kernel on Hyperbolic Space

The purpose of this note is to provide a new proof for the explicit formulas of the heat kernel on hyperbolic space. By definition, the hyperbolic space Hn is a (unique) simply connected complete n-dimensional Riemannian manifold with a constant negative sectional curvature −1. 1991 Mathematics Subject Classification 58G32, 58G11.

On the hyperbolic dirichlet to neumann functional in Hn and Sn

We prove the injectivity of the linearizatíon of the hyperbolíc Dirichlet to Neumann functional in a "small" bounded domain of the hyperbolic space Hn (resp., the sphere Sn) under some suitable transversality condition.

Harmonic Analogues of G. R. Maclane's Universal Functions

Let E denote the space of all entire functions, equipped with the topology of local uniform convergence (the compact-open topology). MacLane [15] constructed an entire function f whose sequence of derivatives (f, f′, f″, …) is dense in E; his construction is succinctly presented in a much later note by Blair and Rubel [2], who unwittingly rederived it (see also [3]). We shall call such a function f a universal entire function. In this note we show that analogous universal functions exist in the space HN of functions harmonic on ℝN, where N⩾2. We also study the permissible growth rates of universal functions in HN and show that the set of all such functions is very large. For purposes of comparison, we first review relevant facts about universal entire functions. The function constructed by MacLane is of exponential type 1. Duyos Ruiz [7] observed that a universal entire function cannot be of exponential type less than 1. G. Herzog [11] refined MacLane's growth estimate by proving the existence of a universal entire function f such that ∣f(z)∣=O(rer) as ∣z∣=r→∞. Finally, Grosse–Erdmann [10] proved the following sharp result.

Positive Eigenfunctions of the Laplacian and Conformal Densities on Homogeneous Trees

In this paper, we study some asymptotic aspects of the positive eigenfunctions of the combinatorial Laplacian associated to a homogeneous tree. The results are inspired by results of Dennis Sullivan concerning λ-harmonic functions on the hyperbolic spaces Hn and contained in the paper [9].

Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent

It is proved that commensurable hyperbolic groups are bi-Lipschitz equivalent. Therefore, subgroups of finite index in an arbitrary hyperbolic group also share this property. In addition, it is shown that any two separated nets Γ1 and Γ2 in the hyperbolic space Hn of dimension n≥2 are bi-Lipschitz-equivalent. These results answer the questions posed in [1].

Strongly Isospectral Hyperbolic and Elliptic Manifolds

In this paper, we prove that two compact hyperbolic manifolds Γ1\\Hn and Γ2\\Hn are strongly isospectral if and only if the regular representations of the full group G of isometries of the hyperbolic space Hn into L2(Γ1\\G) and L2(Γ2\\G) are equivalent. The same result is proved for elliptic manifolds.

A Kernel Theorem on the Space [ H μ × A; B

Recently, we introduced a space [Hfi(A);B] which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119(1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of [Hf, x A ; B], it can be uniquely represented by an element of [H^A) ; B] and hence of [H^ ; [A ; B]]. This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space Dgn x V where F is a Frechet space. His work is based on the facts that the space D$n is an inductive limit space and the convolution product is well defined in Dr . This is not possible here since the space Hn(A) is not an inductive limit space. Furthermore, D(A) is not dense in Hf,(A). To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A .