The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.     

The Horizontal Heat Kernel on the Quaternionic Anti-De Sitter Spaces and Related Twistor Spaces

Potential Analysis - The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel...     

Stability of complex hyperbolic space under curvature-normalized Ricci flow

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties.     

Scott's rigidity theorem for Seifert fibered spaces; revisited

We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).

     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

A Geometric Construction of the Hyperbolic Fibrations Associated with a Flock, q Even

Baker et al. [2] show, via algebraic methods, that a regular hyperbolic fibration of PG(3, q) with constant back half gives rise to a flock of a quadratic cone in PG(3, q), and conversely. In this paper a geometric construction for q even of the flock from the hyperbolic fibration, and conversely, will be described. A proof will be given that this geometric construction indeed corresponds to the known algebraic one. Deirdre Luyckx - The author is Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium) (F.W.O.—Vlaanderen).     

Holomorphic helix of proper order 3 on a complex hyperbolic plane

With the aid of the natural fibration on a complex hyperbolic plane we study holomorphic helices of proper order 3 and show that they are not bounded if their second curvature is sufficiently large.     

Hausdorff Dimension and Limits of Kleinian Groups

In this paper we prove that if M is a compact, hyperbolizable 3-manifold, which is not a handlebody, then the Hausdorff dimension of the limit set is continuous in the strong topology on the space of marked hyperbolic 3-manifolds homotopy equivalent to M. We similarly observe that for any compact hyperbolizable 3-manifold M (including a handlebody), the bottom of the spectrum of the Laplacian gives a continuous function in the strong topology on the space of topologically tame hyperbolic 3-manifolds homotopy equivalent to M. Submitted: January 1998.     

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.     

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.     

The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold

Let M be an n-dimensional noncompact complete Riemannian manifold, "Δ" is the Laplacian of M. It is a negative selfadjoint operator in L²(M). First, we give a criterion of non-existence of eigenvalue by the heat kernel. Applying the criterion yields that the Laplacian on noncompact constant curvature space form has no eigenvalue. Then, we give a geometric condition of M under which the Laplacian of M has eigenvalues. It implies that changing the metric on a compact domain of constant negative curvature space form may yield eigenvalues.     

Spectral stability of the Neumann Laplacian

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category, then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.     

The Homogeneous Geometries of Real Hyperbolic Space

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has two connected components.     

Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC ¹ functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian. Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province (105129) Research supported by Academy of Finland     

Geometry on Probability Spaces

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general. There is one situation where, in the last several decades, the problem has been studied with some success. That is when the underlying space is finite (or even discrete). The introduction of the graph Laplacian has been a major development in algorithm research and is certainly useful for unsupervised learning theory. The approach taken here is to take advantage of both the classical research and the newer graph theoretic ideas to develop geometry on probability spaces. This starts with a space X equipped with a kernel (like a Mercer kernel) which gives a topology and geometry; X is to be equipped as well with a probability measure. The main focus is on a construction of a (normalized) Laplacian, an associated heat equation, diffusion distance, etc. In this setting, the point estimates of calculus are replaced by integral quantities. One thinks of secants rather than tangents. Our main result bounds the error of an empirical approximation to this Laplacian on X.     

On the Geometry of Hyperbolic Surfaces with a Conical Singularity

The geometry of compact hyperbolic surfaces with conical singularities isinvestigated. The Teichmüller space of a hyperbolic pair of pants with asingle singularity is studied. A finite number of real parameters whichdetermine the geometry of a hyperbolic torus of genus two with a singlesingularity, is given.     

A weak version of the Lipman–Zariski conjecture

Markushevich and Tikhomirov provided a construction of an irreducible symplectic V-manifold of dimension 4, the relative compactified Prym variety of a family of curves with involution, which is a Lagrangian fibration with polarization of type (1,2). We give a characterization of the dual Lagrangian fibration. We also identify the moduli space of Lagrangian fibrations of this type and show that the duality defines a rational involution on it.     

On base manifolds of Lagrangian fibrations

We consider base spaces of Lagrangian fibrations from singular symplectic varieties.After defining cohomologically irreducible symplectic varieties,we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space.We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.     

The Fujita exponent for the Cauchy problem in the hyperbolic space

It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.     

Curvatures properties of Lie hypersurfaces in the complex hyperbolic space

A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.