Teichmuller Theory and Low-Dimensional Geometric Variational Problems

AbstractAward：DMS-0505603主要研究者：Michael Wolf这些项目集中于欧几里德三维空间中的完备极小曲面、双曲平面上的调和映射和黎曼曲面上的射影结构等问题。霍夫曼和M. Weber希望证明任意亏格的嵌入螺旋面的存在，在亏格1的存在结果的基础上，他采用了一种不同的方法来简化亏格1的论证。 极小曲面的一个基本的唯一性问题也将被研究：有多少种方法可以使两个平面的相交奇异化？ 主要研究者已经研究了调和映射到双曲planefor几年，导致一个自然的猜想extensionsof准对称的圆双曲盘。 黎曼曲面上的投影结构的计划工作将与博士后研究员大卫大仲马共同进行，目的是为这种结构开发精细的渐近性。 一个稳定的肥皂泡通常占据所有可能的表面中最小的面积，而改变表面必须增加其面积的数学陈述转化为偏微分方程。 由于物理和几何两方面的原因，自19世纪以来，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表面形状像一个螺旋或停车坡道，螺旋面的属一版本可以被描述为一个停车坡道与空气轴-一些有趣的和有吸引力的图片，这些和其他最小的表面可在M。Weber's Web pages，http：//www.indiana/edu/campaignminimal.

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.