Reciprocality of p-modulus and consequences in metric spaces

Abstract

Arne Beurling first studied extremal length, namely the reciprocal of 2−Modulus, in the plane, and then later studied it jointly with Lars Ahlfors. Beurling was interested in extremal length because he wanted a conformal invariant to study harmonic measure. One of the differences between R² and R[superscript N] with N ≥ 3, is that there are far fewer conformal maps in the latter case. This naturally suggests defining a larger class of functions that distort N-Modulus by a bounded amount. This gives rise to the notion of quasiconformal mappings, see [1]. There have been many recent developments in the discrete theory of p-Modulus, and a natural question is “Can the discrete theory tell us anything about the continuous theory?” There are two ways to try and answer this question. The first is to approximate a domain with a mesh of points and study if discrete p-Modulus of families of walks on the mesh converges to continuous p-Modulus on the domain. This line of inquiry has been pursued in the recent literature [10, 11, 21, 25, 12]. The second way to answer the question is to try to come up with a dictionary of results by developing a way to pair up results for the discrete theory and the continuous theory. This is where this thesis is developed. In [6], with Nathan Albin, Pietro Poggi-Corradini, and Nageswari Shanmugalingam, we establish a relationship between ∞-Modulus of a family of paths connecting two points in general metric spaces and the “essential” shortest path metric between two points. This result is inspired by a similar relationship in the discrete setting established in [5]. In [4] N. Albin, Jason Clemens, Nethali Fernando, and P. Poggi-Corradini show that p-Modulus, 1 ≤ p < ∞, can be related to other metrics. Using the work of Aikawa and Ohtsuka, we show that a similar modulus metric can be defined, with some slight modification, in R2, for 2 < p < ∞. Note that, in the continuous setting, with N-dimensional Lebesgue measure, we cannot hope to get a metric for 1 ≤ p ≤ N because for these values the p-Modulus of the family of curves connecting two distinct points is zero. We are currently working to adapt the argument to dimension N ≥ 3 and in metric measure spaces (X,d,µ) where µ is a Borel regular measure.