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

该项目研究几何分析中的优化问题，即在约束下优化能量或面积的结构。存在和结构结果在工程、物理和化学等领域都很重要。经典的例子是最小表面，它在固定的边界条件下使局部面积最小化，例如由各种形状的电线支撑的肥皂膜。本项目研究恒定平均曲率（CMC）和最小曲面以及调和映射。CMC表面优化面积，但由于封闭体积的限制，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，并且与最小表面密切相关。本项目研究的对象在数学的许多分支中都有表征；问题和期望的结果是广泛的兴趣在数学和超越。本研究项目主要研究浸入光滑流形中的CMC曲面和度量空间中的调和映射。在谐波图的研究中，本项目旨在为解决坎农猜想提供一个新的方向。计划建立从圆单位球到具有上曲率界的度规球的调和同胚的存在性。在第二个方向上，该项目旨在改进产生具有上曲率边界的度量空间调和映射的紧致性理论的技术。虽然在这种情况下证明紧性的技术必然是几何和变分的（而不是解析的），但结果与在光滑情况下建立紧性的结果类似。利用改进的技术，研究者计划使用能量而不是连续模量方法建立谐波替代论证。其他研究方向与CMC表面的研究有关。研究人员计划扩展和改进在欧几里得空间中产生CMC超曲面的粘合结构。新的结构有望在欧几里得空间中产生非旋转的环形液滴，并将作为后续结构的模型，在三流形中产生CMC环面。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.