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

AbstractAward：DMS-0071862首席研究员：山田澄夫拟议研究的两个主要课题是亏格大于1的紧致黎曼曲面的Teichmueller空间和泛Teichmueller空间，它们可以用圆的同构群来识别。 在经典和泛Teichmueller空间上都存在一个自然的黎曼度量，称为Weil-Petersson度量。 虽然这些空间不是对称空间，也没有经典李群等距作用于它们（不像具有特殊线性群的双曲空间），但人们注意到，在经典Teichmueller空间上，黎曼曲面的映射类群等距作用，而在普适Teichmueller空间上，圆的反同态群等距作用。 利用Teichmueller空间上的这些群作用，可以构造各种流形上的平坦丛，其纤维是Teichmueller空间的拷贝，这就引出了强/超刚性问题. 正是在这种背景下，研究人员打算使用Teichmueller空间的Weil-Petersson几何。黎曼曲面的主题一直是近两个世纪以来现代数学的中心主题之一。 近年来，由于所谓的超弦理论引起的兴趣，它重新受到关注，物理学家希望用超弦理论构建引力和量子物理学的大统一理论（GMT）。 Teichmueller空间，这是本研究的主题，起着重要的作用，因为它是已知的参数化的“弦的形状。“数学和理论物理之间的相互作用已被证明是富有成效的双方，这是研究人员的信念，拟议的研究可能会提供一些新的方式来理解在相关的科学领域的突出问题。

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.