Teichmuller Theory and Geometric Variational Problems

项目编号：dms -9971563首席研究员：Michael WolfMichael Wolf建议继续研究Teichmuller理论在空间完全极小曲面中的应用，以及奇异空间中调和映射的新技术在Teichmuller理论、离散群和（光滑）调和映射理论中的应用。特别地，他建议将他现有的寻找完全极小曲面的方法扩展到周期极小曲面，然后通过结合teichmuller理论和极小曲面方法证明它们的嵌入性，最后研究这些曲面的族随着其属的增加的极限。他还建议将曲面的最小映射分类到与黎曼曲面上的凸投影结构相关的特定建筑中（为了识别曲面群的离散忠实表示空间的几何自然紧化到凸投影变换的李群中）。最后，他提出利用奇异空间的调和映射（既有温和的奇点，也有更严重的奇点，就像真正的树一样）来研究拟紫红空间贝斯片上的弯曲测度坐标，以及双曲空间之间的调和映射。所有这一切背后的基本问题是，“我们可能遇到的表面的可能形状是什么？”，以及“如果我们要求表面在某种意义上有效地利用材料，会出现什么形状？”当然，“形状”和“效率”这两个词有很多种解释，拟议研究的不同应用很可能涉及对这两个词的不同解释。这项研究试图提高我们对肥皂膜形状的可能性的理解（它有效地利用了材料——其他人将这一理论应用于材料科学尚处于萌芽阶段）以及其他三种形状。我们还研究了以一种节能的方式将一个表面转化为另一个表面的可能方法。事实证明，所有这些不同的问题都是密切相关的，所以一个领域的进步往往会导致其他领域的进步。

AbstractAward: DMS-9971563Principal Investigator: Michael WolfMichael Wolf proposes to continue his studies of the applicationsof Teichmuller theory to the theory of complete minimal surfacesin space and of the applications of new techniques in harmonicmaps to singular spaces to problems in Teichmuller theory,discrete groups and (smooth) harmonic maps theory. Inparticular, he proposes to extend his present methods for findingcomplete minimal surfaces to periodic minimal surfaces, thenproving their embeddedness via a combination ofTeichmuller-theoretic and minimal surface methods, and thenfinally studying limits of families of these surfaces as theirgenus increases. He also proposes to classify the minimal maps ofsurfaces into a particular building associated to convexprojective structures on Riemann surfaces (in order to identify ageometrically natural compactification of the space of discretefaithful representations of a surface group into the Lie group ofconvex projective transformations). Finally, he proposes to useharmonic maps to singular spaces (both with mild singularitiesand more serious singularities, like real trees) to study bendingmeasure coordinates on the Bers slice of Quasifuchsian space, andharmonic maps between hyperbolic spaces.The fundamental questions underlying all of this are, "What arethe possible shapes of surfaces we might encounter?", and "Whatshapes arise if we require the surfaces to be efficient users ofmaterial, in some sense?" There are, of course, many ways ofinterpreting the words "shape" and "efficient", and differentapplications of the proposed research would most likely involvedifferent interpretations of those words. This research projectis an attempt to advance our understanding of the possibilitiesfor the shapes of soap films (which efficiently use material --some applications of this theory by others to material science isin its embryonic stage) and for three other types of shapes. Wealso study the possible ways of transforming one surface intoanother in an energetically efficient way. It turns out that allthese different problems are deeply interrelated, so advances inone area often lead to advances in others.