# 1999 Mt. Baldy Conference on Analysis: Abstracts

### Michael Christ (UC Berkeley): On One-Dimensional Schroedinger Operators with Slowly Decaying Potentials

The spectrum and (generalized) eigenfunctions of a time-independent Schroedinger operator

-d^{2}/dx^{2}+ V(x)

determine the nature of the associated time evolution described by Schroedinger's equation. They depend on the potential in a highly nonlinear manner.

We study the asymptotic behavior of generalized eigenfunctions in
one dimension, for slowly decaying potentials, and show that for
almost every *k* there exists a generalized eigenfunction
behaving roughly like *exp(i k x)* for large *x*. The
eigenfunctions are expressed as sums of infinite series of multilinear
operators applied to the potential, and these series are shown to
converge for almost every energy, with respect to Lebesgue measure,
for a wide class of potentials that is nearly optimal for this
conclusion. In particular, potentials in Lebesgue spaces are
treated.

The proof is related to an ancient theorem of Menshov, Paley, and Zygmund about almost everywhere convergence of Fourier integrals and more general orthogonal expansions. We present a simple and rather general maximal theorem which implies their result, and has other applications. This is joint work with Alexander Kiselev.

### Clifford Earle (Cornell University): Variation of Moduli Under Holomorphic Motions

The theory of holomorphic motions originated as a tool for applying the method of quasiconformal variations to problems in complex dynamics, but it also can be used to study questions in geometric function theory. An important early example is Burt Rodin's 1986 paper “Behavior of the Riemann Mapping Function Under Complex Analytic Deformations of the Domain”.

Recently, in joint projects with Sudeb Mitra and with Adam Epstein, we have used holomorphic motions and quasiconformal variations to answer some questions posed by Dieter Gaier at a 1996 conference on function theory at Oberwolfach. Rodin's theorem can be applied to one of these questions, which concerns the conformal radius of variable slit domains. We shall discuss two of Gaier's questions, with particular emphasis on the conformal radius problem.

### Robert Hardt (Rice University): Limits Associated with Sequences of Smooth Maps to Spheres

Smooth maps between Riemannian manifolds are often not strongly dense in classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 Ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. With T. Riviere, we also treat the case of finite 3-energy maps from the 4 Ball to the 2 sphere. Here the notion of the Hopf degree rather than ordinary topological degree is relevant. The objects produced now may no longer have finite mass, but are nevertheless describable as graphs of atomic measure-valued functions on the reals which are essentially contained in countably 1-rectifiable sets. The crucial compactness of these functions is obtained by a fractional maximal function estimate. There are related associated variational problems for the mappings and for the currents.

### Thomas M. Liggett (UCLA): Growth Models on Trees

I will discuss a system whose state at any time is a random connected subset of the vertices of a homogenous tree. It evolves by adding neighbors at (exponential) rate beta, and deleting leaves at rate 1. There is a phase transition in beta: the system dies out if beta is small, and explodes if beta is large. I will explain how one can compute the critical value for this phase transition exactly. One inequality involves a simple combinatorial argument. The other is harder, and uses a number of analytic and combinatorial tools. One of the former stems from an old (1952) paper of Royden on the existence of Green functions. The talk is based on the following papers:

- A. Puha. A reversible nearest particle system on the homogeneous tree, J. Th. Probab. 12 (1999) 217-254.
- A. Puha. Critical exponents for a reversible nearest particle system on the binary tree, Ann. Probab., to appear.
- T. M. Liggett. Monotonicity of conditional distributions and growth models on trees, to appear.

### Brad Osgood (Stanford University): Recent Results on the Geometry of Univalence Criteria

We will discuss the geometry of univalence criteria associated with the Schwarzian derivative, from the 1949 result of Nehari that started the subject, to recent results on John domains, quasiconformal, and homeomorphic extensions. A natural setting is a generalization of the Schwarzian that allows for conformal changes of metric, and a common theme is convexity, especially convexity in the hyperbolic metric.