A Locally Trivial Quantum Hopf Fibration

The irreducible *-representations of the polynomial algebra of the quantum3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C^*-algebra are shown to coincide with their classical counterparts. The U(1)-action on corresponding for p=1=q to the classical Hopf fibration is proven to be Galois (free). The thus obtained locally trivial Hopf–Galois extension is shown to be equivariantly projective (admitting a strong connection) and non-cleft. The latter is proven by determining an appropriate pairing of cyclic cohomology and K-theory. Presented by S. L. Woronowicz Mathematics Subject Classifications (2000) 16W30, 46L87.     

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.     

The Heat Kernels of the Real Second Classical Domains

L3t[1]istherealsecondclassicaldomains.ThemapdefinedbyW~(I T)--'(I--T)carriesRillontotheunboundeddomainWhichisinvariantfinderthelineartransformationW=ATA',whereAisannxnupertrianglarmatrixwithpositivediagonalelements.SuchtransformationsformaLiegroupRill(n),whichisequivalenttoRillunderthemapdefinedbyW=AA'.TheAiscalledthehoro-hypercirclecoordinateofRRII.Itiswell-knownthatinRillthereisanillvariantmetricdsZ=tr[(I--KW)--'dW]'.InRHI,,whichisequaltothemetricdsZ=fir[(W--'dW)'].Inthehoro-h…     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations.     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations. (Received 30 December 2000; in revised form 11 April 2001)     

The Heat Kernels of a Hypersurfaceof Third Classical Domains

Letisthehypersurfaceofthirdclassicaldomainnlll(Ref.[l]),whereWhichisinvariantunderthelineartransformationW=ATA',AJ=JA,whereAllisannxnuppertriangularmatrixwithpositivediagonalelements,A12isannxnuppertriangularmatrixwithzerodiagonalelements.SuchtransformationsformaLiegroupSglll.whichisequivalenttoHnlllunderthemapdefinedbyW=AAI.TheAiscalledthehorohypercirclecoordinateofHnlll.Itiswell-knownthatinHnlllthereisaninvariantmetricInSnlll,whichisequaltothemetricdsZ~fir[(Z--'dZ)'l,inthehoro-hyp…     

球面上热核的渐近展开

具体计算了球面Ｓｎ（１）（ｎ２）上热核的渐近展开式中前五项的系数，而根据已有的公式只能算出前四项．最后给出了展开式一般项的递推公式，发现它与Ｂｅｒｎｏｕｌｉ数有未曾想到的联系．根据不变量理论，我们可以确定任意ｎ维紧致无边Ｒｉｅｍａｎｎ流形上热核的渐近展开式中第五项的系数．     

关于型III对称空间的热核构造

引进了第III类典型域上的内切超圆坐标;计算出了矩阵极坐标下相应于不变度量的体积元素;利用积分变换构造出了不变度量的Laplace-Beltrami算子的热核.     

Killed Random Processes and Heat Kernels

Let V(x) ≥ 0 be a given function tending to a constant at infinity. It is well known that the density of the Brownian motion B_t killed at the infinitesimal rate V is a Green's function for the heat operator with such a potential. With an appropriate generalization, its Laplace transform also gives the density of ∫ ₀ ^t V(B_s)ds. We construct such a Green's function via spectral analysis of the classical one-dimensional stationary Schrodinger operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 423–432, August, 2005. An erratum to this article is available at .     

Local Lower Bounds on Heat Kernels

We analyze convolution semigroups on a regular measure space which satisfies the local doubling property. We assume the kernels are bounded and symmetric with the characteristic small-time, volume-dependent, singularity. Then, using a weak conservation property, we deduce local lower bounds with a comparable singularity.Applications are given to a wide range of subelliptic and strongly elliptic self-adjoint, or near self-adjoint, operators on Lie groups.     

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...     

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.

     

Heat Flows of Subelliptic Harmonic Maps into Riemannian Manifolds with Nonpositive Curvatures

In this paper, we discuss the heat flows of subelliptic harmonic maps into Riemannian manifolds with nonpositive curvatures, and prove the homotopic existence which is a generalization of the Eells–Sampson theorem.     

Hypoelliptic Heat Kernels on Nilpotent Lie Groups

We study property RD in terms of rapid decay of matrix coefficients. We give new formulations of property RD in terms of a L¹-integrability condition of a Banach representation. Combining this new definition with the existence of cyclic subgroups of exponential growth in non-uniform lattices in semisimple Lie groups, we deduce that there exist matrix coefficients associated to several kinds of quasi-regular representations which satisfy a “non-RD condition” for non-uniform lattices. We obtain also that such coefficients can not satisfy the weak inequality of Harish-Chandra.     

On Aronson's Upper Bounds for Heat Kernels

Let L be a uniformly elliptic operator in divergence form onR^d, and let p(t,x,y) be the fundamental solution to the heatequation for L. A new proof is given of Aronson's upper bound: 2000 Mathematics Subject Classification35J15, 60J60.     

Fibration of Idempotent Measures

Ukrainian Mathematical Journal - We prove that the idempotent barycenter map restricted to the points with nontrivial fibers is a trivial fibration with Hilbert cube fibers whenever it is open.     

Estimates of Heat Kernels with Neumann Boundary Conditions

In this paper, we obtain two-sided estimates on the heat kernels for elliptic operators with singular coefficients equipped with mixed boundary conditions.     

A Sharp Comparison Result Concerning Schrodinger Heat Kernels

Let G₀ and G be the heat kernels of and –V respectively.Under some sharp conditions on V, a proof is given that G(x,t;y,0)/G₀(x,t;y,0)can be bounded from above and below by two positive constants.This largely improves the well-known bounds G(x,t;y,0)/G₀(x,t;y,0) t (0,T]. The result answers an open question of V. Liskevichand Yu. Semenov, in the case where V 0. A sharp global boundin time is also obtained when V 0. 2000 Mathematics SubjectClassification 35K05.