Hopf-Lax type formulas and hypercontractivity

Here we study

Dimension dependent hypercontractivity for Gaussian kernels

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein?CUhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton?CJacobi equation. Hypercontractive bounds on the Ornstein?CUhlenbeck semigroup driven by a non-diffusive Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.     

Higgs bundles over foliation manifolds

In this paper, we consider the stability, semi-stability and canonical metric structures on transverse Higgs bundles over a class of foliation manifolds. Also a transversal Bogomolov inequality is obtained.     

Moduli spaces over manifolds with involutions

Most of the work was carried out under the support of K.C. Wong Education Foundation Ltd.     

Jet bundles as manifolds over algebras

This survey is devoted to the geometry of jet bundles defined by local algebras. Particular attention is given to differentiable manifold structures over algebras in these bundles.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 19, pp. 3–22, 1987.     

Pseudoconvex domains spread over complex homogeneous manifolds

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X = G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.     

On discretizations of invariant foliations over inertial manifolds

Invariant foliations over inertial manifolds of partial differential equations under numerical discretizations are studied. It is proved that the numerical method considered as a discrete dynamical system has C¹-close invariant foliations. The rate of the C¹-convergence is estimated as well.     

Characteristic rank of vector bundles over Stiefel manifolds

The characteristic rank of a vector bundle ξ over a finite connected CW-complex X is by definition the largest integer ${k, 0 \leq k \leq \mathrm{dim}(X)}$ , such that every cohomology class ${x \in H^{j}(X;\mathbb{Z}_2), 0 \leq j \leq k}$ , is a polynomial in the Stiefel–Whitney classes w _i (ξ). In this note we compute the characteristic rank of vector bundles over the Stiefel manifold ${V_k(\mathbb{F}^n), \mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}}$ .     

Orientability of vector bundles over real flag manifolds

We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, which include the tangent bundle and also stable bundles of certain gradient flows. Closed formulas, in terms of roots, are provided.     

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.     

Algebraic models defined over Q of differential manifolds

Here we prove that every compact differential manifold has a smooth algebraic model defined over Q. In dimension 2 we find an algebraic model (may be singular) defined over Q and birational over Q to the projective plane.     

Yang-Mills connections over compact strongly pseudoconvex CR manifolds

Dedicated to Professor Noboru Tanaka in his sixtieth birthday