A Study of Riemann Surface via Weil-Peterson Geometry of Teichmuller Spaces

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AbstractAward: DMS-0071862Principal Investigator: Sumio YamadaThe two main subjects of the proposed research are Teichmuellerspaces of compact Riemann surfaces of genus greater than one andthe universal Teichmueller space, which can be identified withthe diffeomorphism group of the circle. There is a naturalRiemannian metric on both classical and universal Teichmuellerspaces, called the Weil-Petersson metric. Although those spacesare not symmetric spaces, and there are no classical Lie groupsacting on them isometrically (unlike the hyperbolic spaces withthe special linear group), one notes that on the classicalTeichmueller spaces the mapping class group of the Riemannsurfaces acts isometrically, and that on the universalTeichmueller space the diffeomorphism group of the circle actsisometrically. These group actions on Teichmueller spaces can beutilized to construct flat bundles over various manifolds withits fibers being copies of Teichmueller spaces, which leads tothe questions of strong/super rigidity. It is in this contextthat the investigator intends to use the Weil-Petersson geometryof Teichmueller spaces.The subject of Riemann surfaces has been one of the centralthemes of modern mathematics over the last two centuries. Inrecent years, it has received renewed attention due to theinterest created by the so-called super-string theory, with whichphysicists hope to construct the grand unifying theory (GMT) ofgraviation and quantum physics. The Teichmueller space, which isthe subject of this investigation, plays an important role sinceit is known to parametrizes the "shapes of the string."Interactions between mathematics and theoretical physics haveproved to be fruitful for both sides, and it is theinvestigator's belief that the proposed research may offer somenew way of understanding the outstanding issues in the relevantscientific fields.