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

摘要/ abstract摘要：dms -0071862项目负责人：Sumio yamada提出了两个主要的研究对象：属大于1的紧Riemann曲面的Teichmueller空间和可以用圆的微分同构群来识别的泛Teichmueller空间。在经典和通用的teichmueller空间上都有一个自然黎曼度规，称为Weil-Petersson度规。尽管这些空间不是对称空间，并且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，在经典的teichmueller空间上，黎曼曲面的映射类群等距作用，而在一般的teichmueller空间上，圆的微分同态群等距作用。这些在Teichmueller空间上的群作用可以用来在各种流形上构建平面束，其纤维是Teichmueller空间的副本，这就导致了强/超刚性的问题。正是在这种情况下，研究者打算使用Teichmueller空间的Weil-Petersson几何。在过去的两个世纪里，黎曼曲面一直是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，物理学家希望利用超弦理论构建重力航空和量子物理学的大统一理论（GMT）。Teichmueller空间是本次研究的主题，它起着重要的作用，因为已知它可以参数化“弦的形状”。数学和理论物理之间的相互作用已被证明对双方都是富有成效的，研究者相信，所提出的研究可能会为理解相关科学领域的突出问题提供一些新的途径。

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.