Teichmuller Theory and Low-Dimensional Geometric Variational Problems

摘要奖：DMS-0505603主要研究员：Michael Wolf这些项目集中在欧几里德三维空间中的完备性极小曲面、双曲平面上的调和映射以及黎曼曲面上的射影结构等问题上。主要研究员与D.Hoffman和M.Weber合作，希望证明任意亏格的嵌入螺旋面的存在，继续研究它们在亏格1中的存在结果，但寻求一种不同的方法来简化亏格1的论证。还将研究极小曲面上的一个基本唯一性问题：两个平面的交点可以用多少种方法来单一化？首席调查者研究双曲平面上的调和映射已有数年之久，由此引出了关于圆的准对称性向双曲圆盘的扩展的一个自然猜想。黎曼曲面上射影结构的计划工作将与博士后大卫·杜马斯共同进行，目的是为这种结构开发精细的渐近性。极小曲面是横跨金属丝的肥皂泡的数学理想化。一个稳定的肥皂泡通常假定横跨金属丝的所有可能表面的面积最小，而改变表面必然会增加其面积这一事实的数学陈述可以转化为一个偏微分方程式。自19世纪以来，出于物理和几何方面的原因，人们对这些问题的各种版本进行了深入的研究。上面提到的螺旋面是一个形状像开瓶器或停车坡道的表面，螺旋面的亏格可以描述为没有通风井的停车坡道--这些和其他极小曲面的一些有趣和吸引人的照片可以在M.Weber的网页上找到，http://www.indiana/edu/~minimal.

AbstractAward: DMS-0505603Principal Investigator: Michael WolfThese projects concentrate on problems concerning completeminimal surfaces in Euclidean three-space, harmonic maps onto thehyperbolic plane, and projective structures on Riemann surfaces.The principal investigator, working with D. Hoffman and M. Weber,hopes to prove the existence of embedded helicoids of arbitrarygenus, following up on their existence result in genus one butpursuing a different approach that is already known to simplifythe genus-one argument. A basic uniqueness question on minimalsurfaces is also going to be studied: In how many ways can onedesingularize the intersection of two planes? The principalinvestigator has studied harmonic maps onto the hyperbolic planefor several years, leading to a natural conjecture on extensionsof quasi-symmetries of the circle to the hyperbolic disk. Theplanned work on projective structures on Riemann surfaces will bepursued jointly with a postdoctoral fellow, David Dumas, and aimsto develop fine asymptotics for such structures.A minimal surface is the mathematical idealization of a soapbubble spanning a wire. A stable soap bubble ordinarily assumesthe least area of all possible surfaces spanning that wire, andthe mathematical statement of the fact that varying the surfacemust increase its area translates into a partial differentialequation. Versions of these problems have been studied intenselysince the 19th century and before, for both physical andgeometric reasons. The helicoid referred to above is a surfaceshaped like a corkscrew or parking ramp, and the genus-oneversion of the helicoid could be described as a parking ramp withan airshaft -- some interesting and appealing pictures of theseand other minimal surfaces are available on M. Weber's Web pages,http://www.indiana/edu/~minimal.