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

本项目研究几何分析中的优化问题，即在约束条件下优化能量或面积的构造。存在性和结构性结果在工程、物理和化学等领域都很有意义。经典的例子是极小曲面，它局部地最小化受固定边界条件约束的面积，例如由各种形状的线支撑的肥皂膜。这个项目研究常平均曲率（CMC）和极小曲面以及调和映射。CMC表面优化面积，但受到封闭体积的限制- CMC表面在自然界中看起来像肥皂泡。调和映射优化能量而不是面积，并且与极小曲面密切相关。本研究课题的研究对象在数学的许多分支中都具有特征，所提出的问题和期望的结果在数学界及其他领域都具有广泛的意义。本研究课题主要研究浸入光滑流形的CMC曲面和度量空间的调和映射。在调和映射方面的工作，该项目旨在为解决Cannon猜想提供一个新的方向。计划建立从圆形单位球面到具有曲率上界的度量的球面的调和同胚的存在性。在第二个方向，该项目的目的是完善技术，产生了一个紧性理论的调和映射到度量空间的曲率上界。虽然在这种情况下证明紧性的技术必须是几何的和变分的（而不是分析的），但结果类似于在光滑环境下建立紧性的技术。使用完善的技术，调查人员计划建立一个谐波替换参数使用能量，而不是连续性方法的模量。其他研究方向涉及CMC表面的研究。研究人员计划扩展和改进一种在欧几里得空间中产生CMC超曲面的胶合结构。新的建设预计将产生非旋转，在欧几里得空间的环形下降，并将作为一个模型，为后续建设，以生产CMC环面在three-manifold.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.