We show that a fi-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fi-Gaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

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The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination A and the associated hyperbolic 3-lamination H endowed with an action of a discrete group of isomorphisms.

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The scientific aim of computational neuroanatomy using magnetic resonance imaging (MRI) is to quantify inter- and intra-subject morphological variabilities. A unified statistical frame- work for analyzing temporally varying brain morphology is presented. Based on the math- ematical framework of differential geometry, the deformation of the brain is modeled and key morphological descriptors such as length, area, volume dilatation and curvature change are computed. To increase the signal-to-noise ratio, Gaussian kernel smoothing is applied to 3D images. For 2D curved cortical surface, diffusion smoothing, which generalizes Gaus- sian kernel smoothing, has been developed. Afterwards, statistical inference is based on the excursion probability of random fields defined on manifolds.

Contents 1 Introduction 2 2 Preliminaries 2 2.1 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 The doubling property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 The Poincar'e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 The p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 The non-smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 p-parabolicity and p-hyperbolicity 10 3.1 An inequality for supersolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Volume growth and p-parabolicity . . . . . . . . . . . . . . . . .

Abstract. The main result of this paper is that a connected bounded geometry complete Kähler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not Kähler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact Kähler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2. This version of this paper contains details not in the version submitted for publication.

this paper do not directly use the Ricci curvature assumption. In fact, we will show that Theorems 1.1, 1.2 hold true for any manifold which is quasi-isometric (even roughly-isometric, under any reasonable bounded local geometry assumption) to a manifold with non-negative Ricci-curvature. In particular, the bounds of Examples 1.1, 1.2, hold true if the Laplace operator is replaced by a uniformly elliptic operator in divergence form. Thus, the bounds stated above are reasonably stable. The adequate hypothesis for our purpose is expressed in terms of a parabolic Harnack inequality or, equivalently, in terms of certain Poincar'e inequality and volume growth (see below Section 2.2). The present work originated from our desire to understand the behavior of the heat kernel on manifolds with more than one ends. Indeed, together with good estimates of certain hitting probabilities obtained in [14], the result presented here is one of the main building blocks in the proof of the sharp estimates for the heat kernel on manifolds with ends that have been announced in [11] and are proved in [12]. The following result complements Theorems 1.1, 1.2 in this direction. Given a Riemannian manifold with k ends, let U be a relatively compact open set in M with smooth boundary such that M n U has exactly k unbounded connected components E 1 ; : : : ; E k . Let K i = @U " E i , and consider E i as a manifold with boundary ffi E i := K i . Denote by p i the heat kernel on E i and by p\Omega i the Dirichlet heat kernel on\Omega i = E i n K i (in other words, p i satisfies the Neumann condition on K i , whereas p\Omega i satisfies the Dirichlet condition on K i ). Let also V i (x; t) be the volume function on E i . For each end E i , fix a point o i 2 K i and define the functions H i , D...

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