Teichmuller Theory and Low-Dimensional Geometric Variational Problems

项目主要研究欧几里得三维空间中的极小曲面、双曲平面上的调和映射、黎曼曲面上的投影结构等问题。首席研究员，与D. Hoffman和M. Weber合作，希望证明任意属的嵌入螺旋体的存在，继续它们的存在，得到属一，但追求一种不同的方法，这种方法已经知道可以简化属一的论点。最小曲面上的一个基本唯一性问题也将被研究：有多少种方法可以将两个平面的交点分离？这位首席研究员研究双曲平面上的调和映射已经好几年了，由此产生了一个关于圆的准对称性向双曲盘的延伸的自然猜想。黎曼曲面上投影结构的计划工作将与博士后研究员David Dumas共同进行，旨在开发这种结构的精细渐近性。最小表面是横跨导线的肥皂泡的数学理想。一个稳定的肥皂泡通常假设在所有可能的表面跨越电线的面积最小，而改变表面必须增加其面积这一事实的数学陈述可以转化为偏微分方程。这些问题的不同版本从19世纪或更早的时候就因为物理和几何的原因而被深入研究。上面提到的螺旋面是一种形状像开瓶器或停车坡道的表面，螺旋面的广义转换可以被描述为带有通风井的停车坡道——这些和其他最小表面的一些有趣和吸引人的图片可以在韦伯的网页上找到，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.