Mathematical Sciences: Harmonic Measure on Riemann Surfaces

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Work on this project relates investigations into properties of analytic functions with the geometry of Riemann surfaces. While this research continues much of the same theme represented in earlier work, the introduction of the tools of Brownian motion add a new dimension to it. The underlying approach to problems addressed here is one of constructing Riemann surfaces which must be image surfaces of analytic functions to ensure that the mappings have some prescribed properties, especially of a geometric nature. An example of the type of application possible is that of determining and locating harmonic measure by using exit times of Brownian motion. If harmonic measure exists, it is supported on accessible boundary points. On the other hand, one constructs analytic functions by building Riemann surfaces which project onto the plane and then combine with a Riemann mapping function. The difficult part of three constructions is to verify that the resulting functions have certain properties. Here Brownian techniques are expected to be of value. Particular problems to be addressed deal with the scope of the support of harmonic measure. To what extent can it diffuse rather than concentrate? It has been shown recently that such measures (on Riemann surfaces) can be absolutely continuous with respect to area. A second class of applications involve the identification of singular factors in inner functions. Except for the presence of omitted values, very few criteria exist which detect these factors. Using new geometric constructs, efforts will be made to ensure that the resulting functions have prescribed singular factors. It should be noted that Brownian techniques are not sensitive enough to detect such terms.