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.

Abstractaward：DMS-0505603原理研究者：迈克尔·沃尔夫（Michael Wolf）该项目集中于有关欧几里得三个空间中零件表面的问题在它们的存在上导致了一个属，但出现了一种已知的不同方法，该方法已经简化了属的论点。 还将研究关于最小值的一个基本唯一性问题：以几种方式可以使两架飞机的交集进行多种方式？ 几年来，主要的分门物研究器已经研究了双曲线平面上的谐波图，从而在圆形圆盘上的准隔符上的伸展上进行了自然猜想。 关于Riemann表面上投影结构的计划的工作将与博士后研究员David Dumas共同进行，而Aimsto为此类结构开发了精细的渐近物。 稳定的肥皂泡通常假设所有可能的表面的最小面积跨越了这条线，而数学陈述是变化的表面效果会增加其面积，这转化为部分差异化。 出于物理和几何原因，这些问题的版本已在19世纪及以前进行了深入研究。 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.