Mathematical Sciences: Harmonic Measure on Riemann Surfaces

该项目的工作涉及调查财产 解析函数与黎曼曲面的几何关系。 虽然这项研究继续大部分相同的主题代表 在早期的工作中，布朗运动工具的引入 为它增加了一个新的维度。解决问题的基本方法 这里讨论的是一个构造黎曼曲面，必须 是解析函数的图像表面，以确保 映射有一些规定的属性，特别是 几何性质 可能的应用类型的一个例子是 利用谐波的出射时间确定和定位谐波测度 布朗运动 如果调和测度存在，则它在 可访问的边界点。 另一方面， 解析函数通过建立投影的黎曼曲面 然后联合收割机与黎曼映射函数相结合。 三种构造的难点在于验证 结果函数具有某些属性。 这里布朗运动 技术是有价值的。 需要解决的具体问题涉及到 调和测度的支持。 它能扩散到什么程度 而不是集中精力？ 最近的研究表明， 测度（在黎曼曲面上）可以绝对连续， 尊重地区。 第二类应用涉及识别 内部函数中的奇异因子 除了存在 省略的值，很少有标准存在，检测这些 因素 使用新的几何结构，将努力 确保所产生的函数具有指定的奇异性 因素 应该注意的是，布朗技术不是 敏感到足以检测这些术语。

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.