We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds etc. which are related to recurrence and non-explosion.

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Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapid introduction to techniques in complex dynamics. Its central concern is the structure of an infinitely renormalizable quadratic polynomial f(z) = z2 + c. As discovered by Feigenbaum, such a mapping exhibits a repetition of form at infinitely many scales. Drawing on universal estimates in hyperbolic geometry, this work gives an analysis of the limiting forms that can occur and develops a rigidity criterion for the polynomial f. This criterion supports general conjectures about the behavior of rational maps and the structure of the Mandelbrot set.The course of the main argument entails many facets of modern complex dynamics. Included are foundational results in geometric function theory, quasiconformal mappings, and hyperbolic geometry. Most of the tools are discussed in the setting of general polynomials and rational maps.

https://abel.math.harvard.edu/~ctm/papers/home/text/papers/real/book.pdf

Complex Dynamics and Renormalization

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d ≤ 4. In the present work, we extend Pemantle's alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following: ○ The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant. ○ The tail σ-fields of the WSF and the FSF are trivial on any graph. ○ On any Cayley graph that is not a finite extension of Z, all component trees of the WSF have one end; this is new in Z d for d > 5. . On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. ○ The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space H d is analyzed. . A Cayley graph is amenable iff for all e > 0, the union of the WSF and Bernoulli percolation with parameter e is connected. ○ Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees. We also present numerous open problems and conjectures.

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Uniform spanning forests

/pdf/dynamics-of-quadratic-polynomials-i-ii-10ox00yya4.pdf

Dynamics of quadratic polynomials, I–II

Classification Theory of Riemann Surfaces

Classification Theory of Riemannian Manifolds

/pdf/existence-of-dirichlet-finite-biharmonic-functions-on-the-gdt66wl4d8.pdf

Existence of Dirichlet finite biharmonic functions on the Poincaré 3-ball

Albhough the existence of nonharmonie biharmonic functions on a noncompact Riemannian manifold is always assured, the same is not necessarily true if various boundedness properties are imposed upon them. For this reason it may be interesting to find conditions for the existence of nonharmonie biharmonic functions with given boundedness properties and to compare these conditions for various such properties. As a first step of this biharmonic classification problem we studied in Nakai-Sario [6] (see also O'Malla [8], Nakai [4]) the existence of Dirichlet finite biharmonic functions and showed, among other things, that this existence has no relation to the parabolicity or hyperbolicity of the manifold. In the present paper we will discuss the existence of bounded nonharmonie biharmonic functions. As in the case of Dirichlet finite biharmonic functions, the existence or nonexistence of bounded nonharmonie biharmonic functions turns out to be unrelated to the parabolicity or hyperbolicity of the base manifold. We also show that the existence of Dirichlet finite biharmonic functions has no relation to that of bounded nonharmonie biharmonic functions. In particular the former does not necessarily imply the existence of Dirichlet finite bounded nonharmonie biharmonic functions. Therefore, s is to be expected, there is no analogy with the case of harmonic functions.

Existence of bounded biharmonic functions.

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Leo Sario

Papers

Classification Theory of Riemann Surfaces

Classification Theory of Riemannian Manifolds

Existence of Dirichlet finite biharmonic functions on the Poincaré 3-ball

Existence of bounded biharmonic functions.

Convergence of normal operators