The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ?²ⁿ⁺¹ which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ?²ⁿ⁺¹.     

Integral representation of the sub-elliptic heat kernel on the complex anti-de Sitter fibration

We derive an integral representation for the subelliptic heat kernel of the complex anti-de Sitter fibration. Our proof is different from the one used in Wang (Potential Anal 45:635–653, 2016) since it appeals to the commutativity of the D’Alembertian and of the Laplacian acting on the vertical variable rather than the analytic continuation of the heat semigroup of the real hyperbolic space. Our approach also sheds the light on the connection between the sub-Laplacian of the above fibration and the so-called generalized Maass Laplacian, and on the role played by the odd dimensional real hyperbolic space.     

H型群上$p$-次Laplace算子的Hopf型引理和强极大值原理

本文首先推广了Capogna,Danielli和Garofalo关于p-次Laplace算子的径向解的一个重要公式,然后通过改进欧氏空间中证明Laplace算子的Hopf引理的方法,证明了H型群上p-次Laplace算子的Hopf型引理,进而证明了一个强极大值原理。     

The subelliptic heat kernel on the CR sphere

We study the heat kernel of the sub-Laplacian $L$ on the CR sphere $\mathbb{S }^{2n+1}$ . An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian $-L+n^2$ that was obtained by Geller (J Differ Geom 15:417–435, 1980), and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration $\mathbb{S }^{2n+1} \rightarrow \mathbb{CP }^n$ .     

Harmonicity and minimality of oriented distributions

We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical for the functionals related to the volume or the energy of the map. We show that the three-dimensional distribution ofS ^4m+3 tangent to the quaternionic Hopf fibration defines a harmonic map and a minimal immersion and we extend these results to more general situations coming from 3-Sasakian and quaternionic geometry. Partially supported by DGI Grant No. BFM2001-3548.     

Carnot Geometry and the Resolvent of the Sub-Laplacian for the Heisenberg Group

Abstract

We obtain an explicit representation formula for the sub-Laplacian on the isotropic, three-dimensional Heisenberg group. Using the formula we obtain themeromorphic continuation of the resolvent to the logarithmic plane, the existence of boundary values in the continuous spectrum, and semiclassical asymptotics of the resolvent kernel. The asymptotic formulas show the contribution of each Hamiltonian path in Carnot geometry to the spatial and high-energy asymptotics of the resolvent (convolution) kernel for the sub-Laplacian.     

Substantial Riemannian submersions ofS 15 with 7-dimensional fibres

In this paper we show that a substantial Riemannian submersion ofS ¹⁵ with 7-dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of the diameter rigidity theorem for the Cayley plane which is considerably stronger than the very recent radius rigidity theorem of Wilhelm.     

Milnor fibration at infinity for mixed polynomials

We study the existence of Milnor fibration on a big enough sphere at infinity for a mixed polynomial f: ?²ⁿ → ?². By using strongly non-degenerate condition, we prove a counterpart of Némethi and Zaharia’s fibration theorem. In particular, we obtain a global version of Oka’s fibration theorem for strongly non-degenerate and convenient mixed polynomials.     

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:

• Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
• Volume estimates of metric balls;
• Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
• Spectral gap estimates.

     

FUNCTORIAL PROPERTIES OF THE HOPF INVARIANT I

Abstract

Homotopy operations Θ: [ΣY, U] → [ΣY, V] which are natural in Y are considered. In particular a technique used in the definition of the Hopf invariant (as treated by Berstein-Hilton) shows that any fibration p: E → B with fiber V, when provided with a homotopy section of Ωp, determines such a homotopy operation [ΣY, E] → [ΣY, V]. More generally, starting from a track class of homotopies α º f ? β º g we adapt this fibration technique to construct a homotopy operation [ΣY, M(f,g)] → [ΣY, F _α * F _β] called a Hopf invariant. The intervening fibration in the definition of this Hopf invariant arises via the fiberwise join construction.     

Large time estimates for heat kernels in nilpotent Lie groups

The aim of this paper is to obtain some estimate for large time for the Heat kernel corresponding to a sub-Laplacian with drift term on a nilpotent Lie group. We also obtain a uniform Harnack inequality for a “bounded” family of sub-Laplacians with drift in the first commutator of the Lie algebra of the nilpotent group.     

Elements of Potential Theory on Carnot Groups

We propose and study elements of potential theory for the sub-Laplacian on homogeneous Carnot groups. In particular, we show the continuity of the single-layer potential and establish Plemelj-type jump relations for the double-layer potential. As a consequence, we derive a formula for the trace on smooth surfaces of the Newton potential for the sub-Laplacian. Using this, we construct a sub-Laplacian version of Kac’s boundary value problem.     

On fundamental solution for powers of the sub-Laplacian on the Heisenberg group

We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and to construct its fundamental solution. Besides, the series representation of the fundamental solution for square of the sub-Laplacian on the Heisenberg group is given and we also get the closed form of the fundamental solution for square of the sub-Laplacian on the Heisenberg group with dimension n=2,3,4.     

ISOPARAMETRIC HYPERSURFACES IN CP~n WITH CONSTANT PRINCIPAL CURVATURES

This paper proves that the number of distinct principal curvatures of a realisoparametric hypersurface in CP~n with constant principal curvatures can be only 2, 3 or 5.The prehnage of such hypersurface under the Hopf fibration is an isoparametrichypersarface in S~(2n+l) with 2 or 4 distinct principal curvatures. For real isoparametrichypersurfaces in CP~n with 5 distinct constant principal curvatures a local structuretheorem is given.     

SOME SHARP RELLICH TYPE INEQUALITIES ON NILPOTENT GROUPS AND APPLICATION

We prove some Rellich type inequalities for the sub-Laplacian on Carnot nilpotent groups.Using the same method,we obtain some analogous inequalities for the Heisenberg-Greiner operators.In most cases,the constants we obtained are optimal.     

M-纤维式纤维化的特征与诱导M-纤维式纤维化

本文研究了M-纤维式纤维化的问题,利用M-纤维式升腾函数获得了M-纤维式纤维化的特征,以及证明了M-纤维式上纤维化诱导的映射空间之间的M-纤维式映射在一定条件下也是M-纤维式纤维化.     

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.     

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...