CAREER: Existence and Regularity of Solutions to Variational Problems in Geometric Analysis

职业：几何分析中变分问题解的存在性和规律性

• 批准号: 2147439
• 负责人: Christine Breiner
• 金额: $ 40.12万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2147439&HistoricalAwards=false
• 关键词: CAREER; Existence; Regularity; Solutions; Variational

This project studies optimization questions in geometric analysis, namely constructs that optimize energy or area subject to a constraint. Existence and structural results are of interest in areas such as engineering, physics, and chemistry. Classical examples are minimal surfaces, which locally minimize area subject to fixed boundary conditions, such as soap films supported by wires of various shapes. This project studies constant mean curvature (CMC) and minimal surfaces as well as harmonic maps. CMC surfaces optimize area, but with constraint given by enclosed volume -- CMC surfaces appear in nature as soap bubbles. Harmonic maps optimize energy rather than area and are closely related to minimal surfaces. The objects studied in this project have characterizations in many branches of mathematics; the questions and desired results are of broad interest in mathematics and beyond.This research project primarily studies CMC surfaces immersed in smooth manifolds and harmonic maps into metric spaces. In the work on harmonic maps, the project aims to provide a new direction for resolution of Cannon's conjecture. It is planned to establish the existence of a harmonic homeomorphism from the round unit sphere into a sphere with a metric possessing upper curvature bounds. In a second direction, the project aims to refine techniques that produced a compactness theory for harmonic maps into metric spaces with upper curvature bounds. While the techniques for proving compactness in this setting are necessarily geometric and variational (rather than analytic), the results are analogous to those that establish compactness in the smooth setting. Using the refined techniques, the investigator plans to establish a harmonic replacement argument using energy rather than modulus of continuity methods. Other research directions relate to the study of CMC surfaces. The investigator plans to extend and refine a gluing construction that produced CMC hypersurfaces in Euclidean space. The new construction is expected to produce non-rotational, toroidal drops in Euclidean space and will serve as a model for a subsequent construction to produce CMC tori in three-manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目研究几何分析中的优化问题，即优化受约束的能量或面积的构造。存在性和结构结果在工程、物理和化学等领域引起人们的兴趣。经典的例子是最小表面，它局部最小化受固定边界条件影响的区域，例如由各种形状的线支撑的肥皂膜。该项目研究恒定平均曲率（CMC）和最小曲面以及调和图。 CMC 表面优化了面积，但受到封闭体积的限制——CMC 表面在自然界中表现为肥皂泡。谐波图优化能量而不是面积，并且与最小曲面密切相关。该项目研究的对象具有许多数学分支的特征；这些问题和期望的结果在数学及其他领域引起了广泛的兴趣。该研究项目主要研究沉浸在光滑流形中的 CMC 表面以及度量空间的调和映射。在调和图的工作中，该项目旨在为坎农猜想的解决提供新的方向。计划建立从圆形单位球体到具有曲率上限的度量的球体的调和同胚的存在性。在第二个方向上，该项目旨在改进将调和映射的紧致性理论生成到具有上曲率界限的度量空间的技术。虽然在这种情况下证明紧致性的技术必然是几何和变分的（而不是解析的），但结果与在平滑设置中建立紧致性的技术类似。利用改进的技术，研究人员计划使用能量而不是连续性模量方法来建立谐波替换论证。其他研究方向涉及 CMC 表面的研究。研究人员计划扩展和完善在欧几里得空间中产生 CMC 超曲面的粘合结构。新建筑预计将在欧几里得空间中产生非旋转环形水滴，并将作为后续建筑生产三歧管 CMC 托里的模型。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。

项目成果

Harmonic branched coverings and uniformization of CAT( k ) spheres

CAT( k ) 球体的谐波分支覆盖和均匀化

• DOI: 10.1515/crelle-2021-0031
• 发表时间: 2021-08
• 期刊: Journal für die reine und angewandte Mathematik (Crelles Journal
• 作者: Breiner, Christine;Mese, Chikako
• 通讯作者: Mese, Chikako