Teichmuller Theory and Geometric Variational Problems

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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.