Applications of some elliptic equations in Riemannian manifolds

Let (Mⁿ⁺¹,g)$(M^{n+1},g)$ be a compact Riemannian manifold with smooth boundary B and nonnegative Bakry–Emery Ricci curvature. In this paper, we use the solvability of some elliptic equations to prove some estimates of the weighted mean curvature and some related rigidity theorems. As their applications, we obtain some lower bound estimate of the first nonzero eigenvalue of the drifting Laplacian acting on functions on B and some corresponding rigidity theorems.     

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting.It? ’s formula in terms of the extended stochastic integrals is obtained.     

(Quasi-)regular Dirichlet forms for particle systems on Polish spaces

Explicit sufficient and necessary conditions are presented for reaction-diffusion type Dirichlet forms on Polish spaces studied by Röckner and Wang [Potential Anal., 2006, 24: 223–243] to be (quasi-)regular. As preparations, the (quasi-)regularity of the sum of two or countably many Dirichlet forms is investigated.     

Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.     

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.     

On the geometry of orthonormal frame bundles II

We study the geometry of orthonormal frame bundles OM over Riemannian manifolds (M, g). The former are equipped with some modifications of the Sasaki-Mok metric depending on one real parameter c ≠ 0. The metrics are “strongly invariant” in some special sense. In particular, we consider the case when (M, g) is a space of constant sectional curvature K. Then, for dim M > 2, we find always, among the metrics , two strongly invariant Einstein metrics on OM which are Riemannian for K > 0 and pseudo-Riemannian for K < 0. At least one of them is not locally symmetric. We also find, for dim M ≥ 2, two invariant metrics with vanishing scalar curvature.      

On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets

We obtain a criterion for the quasi-regularity of generalized (non-sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. The second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels.     

Eigenfunctions of the weighted Laplacian and a vanishing theorem on gradient steady Ricci soliton

The aim of this note has two folds. First, we show a gradient estimate of the higher eigenfunctions of the weighted Laplacian on smooth metric measure spaces. In the second part, we consider a gradient steady Ricci soliton and prove that there exists a positive constant c(n)$c(n)$ depending only on the dimension n   of the soliton such that there is no nontrivial harmonic 1-form (hence harmonic function) which is in L^p$L^p$ on such a soliton for any 2<p<c(n)$2<p<c(n)$.     

Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees

Transient random walk on a tree induces a Dirichlet form on its Martin boundary, which is the Cantor set. The procedure of the inducement is analogous to that of the Douglas integral on S¹ associated with the Brownian motion on the unit disk. In this paper, those Dirichlet forms on the Cantor set induced by random walks on trees are investigated. Explicit expressions of the hitting distribution (harmonic measure) ν and the induced Dirichlet form on the Cantor set are given in terms of the effective resistances. An intrinsic metric on the Cantor set associated with the random walk is constructed. Under the volume doubling property of ν with respect to the intrinsic metric, asymptotic behaviors of the heat kernel, the jump kernel and moments of displacements of the process associated with the induced Dirichlet form are obtained. Furthermore, relation to the noncommutative Riemannian geometry is discussed.     

Mass Transport and Variants of the Logarithmic Sobolev Inequality

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting. The research of A.V. Kolesnikov was supported by RFBR 07-01-00536, DFG Grant 436 RUS 113/343/0 and GFEN 06-01-39003.     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions,II

We obtain upper and lower bounds for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 606–621, October–December, 2006.     

Monotone Approximation of Energy Functionals for Mappings into Metric Spaces -- II

We investigate approximations E(f) of energy functionals E(f) for generalized harmonic maps f:MN between singular spaces. Given any symmetric submarkovian semigroup (P) on any measure space (M, ,m) and any metric space (N,d) we study the approximated energy functionals

Convergence of Riemannian Manifolds and Laplace Operators, II

We study spectral convergence of compact Riemannian manifolds or more generally certain Dirichlet spaces, obtaining some compactness results on harmonic functions and harmonic maps. Mathematics Subject Classifications (2000) 53C21, 58D17, 58J50. Atsushi Kasue: Partly supported by the Grant-in-Aid for Scientific Research (B) No. 15340053 of the Japan Society for the Promotion of Science.     

Weak convergence of Dirichlet processes

Weak convergence of Markov processes is studied by means of Dirichlet forms and two theorems for weak convergence of Hunt processes on general metric spaces are established. As applications, examples for weak conver gence of symmetric or non-symmetric Dirichlet processes on finite and infinite spaces are given. Project partially supported by the National Natural Science Foundation of China and Tianyuan Mathematics Foundation.