Teichmuller Theory and Low-Dimensional Geometric Variational Problems

Teichmuller 理论和低维几何变分问题

• 批准号: 0505603
• 负责人: Michael Wolf
• 金额: --
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505603&HistoricalAwards=false
• 关键词: Teichmuller; Theory; Low; Dimensional; Geometric

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.

摘要奖：DMS-0505603 首席研究员：Michael Wolf 这些项目集中于有关欧几里得三空间中的完全最小曲面、双曲平面上的调和映射以及黎曼曲面上的射影结构的问题。首席研究员与 D. Hoffman 和 M. Weber 合作，希望为了证明任意属的嵌入式螺旋面的存在，跟踪它们的存在导致属一，但采用不同的方法，即已知可以简化属一论证。 还将研究最小曲面的基本唯一性问题：可以通过多少种方式对两个平面的交点进行去奇异化？ 首席研究员研究了双曲平面上的调和映射数年，得出了圆的拟对称性扩展到双曲盘的自然猜想。 黎曼曲面上的射影结构的计划工作将与博士后研究员 David Dumas 联合进行，旨在为此类结构开发精细的渐近函数。最小曲面是跨越电线的肥皂泡的数学理想化。 稳定的肥皂泡通常假定跨越该线的所有可能表面的最小面积，并且改变表面必须增加其面积这一事实的数学表述转化为偏微分方程。 自 19 世纪及之前以来，出于物理和几何原因，人们对这些问题的各个版本进行了深入研究。 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，螺旋面的属一版本可以被描述为带有气井的停车坡道——这些和其他最小表面的一些有趣和吸引人的图片可以在 M. Weber 上找到。网页，http://www.indiana/edu/~minimal。