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

基本信息

  • 批准号:
    1750254
  • 负责人:
  • 金额:
    $ 40.12万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2018
  • 资助国家:
    美国
  • 起止时间:
    2018-08-01 至 2021-09-30
  • 项目状态:
    已结题

项目摘要

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.
这个项目研究几何分析中的优化问题,即优化能量或受约束的区域的结构。存在和结构结果是工程学、物理学和化学等领域的研究热点。经典的例子是极小曲面,它在固定边界条件下局部最小化面积,例如由各种形状的导线支撑的肥皂膜。该项目研究常平均曲率(CMC)和极小曲面以及调和映射。CMC曲面优化了面积,但受封闭体积的约束--CMC曲面本质上显示为肥皂泡。调和贴图优化能量而不是面积,并且与最小曲面密切相关。本课题所研究的对象具有多个数学分支的特征,所提出的问题和期望的结果在数学上具有广泛的研究意义。本课题主要研究光滑流形中的CMC曲面和度量空间中的调和映射。在调和映射的工作中,该项目旨在为解决Cannon猜想提供一个新的方向。证明了从圆形单位球面到度规具有曲率上界的球面的调和同胚的存在性。在第二个方向上,该项目的目标是改进产生调和映射紧性理论的技术,使之成为具有曲率上界的度量空间。虽然在这种情况下证明紧致性的技术必须是几何和变分的(而不是解析的),但结果类似于在光滑环境中建立紧致性的那些结果。利用改进的技术,研究人员计划使用能量方法而不是连续模数方法来建立谐波替代论证。其他研究方向与CMC曲面的研究有关。这位研究人员计划扩展和改进一种在欧几里德空间中产生CMC超曲面的粘合结构。新的建筑预计将在欧几里得空间产生非旋转的环状水滴,并将作为后续建筑的模型,以产生三个流形的CMC Tori。这一奖项反映了NSF的法定使命,并通过使用基金会的智力优势和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

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Christine Breiner其他文献

A variational characterization of the catenoid
悬链线的变分表征
Federal Reserve Bank of New York Staff Reports Inflation Risk and the Cross Section of Stock Returns Inflation Risk and the Cross Section of Stock Returns
纽约联邦储备银行工作人员报告通胀风险和股票收益横截面 通胀风险和股票收益横截面
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Fernando M. Duarte;Hengjie Ai;Christine Breiner;D. Cesarini;Hui Chen;Maya Eden;Xavier Gabaix;Jonathan Goldberg;Jennifer La 'o;Guido Lorenzoni;Gustavo Manso;M. Mestieri;Matt Notowididgo;Sahar Parsa;Michael Powell;Jenny Simon;Alp Simsek;Ivo Welch
  • 通讯作者:
    Ivo Welch
Helicoid-like minimal disks and uniqueness
类螺旋最小圆盘及其独特性
  • DOI:
  • 发表时间:
    2008
  • 期刊:
  • 影响因子:
    0
  • 作者:
    J. Bernstein;Christine Breiner
  • 通讯作者:
    Christine Breiner
Distortions of the helicoid
  • DOI:
    10.1007/s10711-008-9290-9
  • 发表时间:
    2008-09-24
  • 期刊:
  • 影响因子:
    0.500
  • 作者:
    Jacob Bernstein;Christine Breiner
  • 通讯作者:
    Christine Breiner
Conservation Laws and Gluing Constructions for Constant Mean Curvature (Hyper)Surfaces
恒定平均曲率(超)表面的守恒定律和粘合结构

Christine Breiner的其他文献

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{{ truncateString('Christine Breiner', 18)}}的其他基金

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
  • 财政年份:
    2021
  • 资助金额:
    $ 40.12万
  • 项目类别:
    Continuing Grant
Existence and Regularity for Variational Problems
变分问题的存在性和正则性
  • 批准号:
    1609198
  • 财政年份:
    2016
  • 资助金额:
    $ 40.12万
  • 项目类别:
    Standard Grant
The local and global structure of variational solutions
变分解的局部和全局结构
  • 批准号:
    1308420
  • 财政年份:
    2013
  • 资助金额:
    $ 40.12万
  • 项目类别:
    Standard Grant
PostDoctoral Research Fellowship
博士后研究奖学金
  • 批准号:
    0902718
  • 财政年份:
    2009
  • 资助金额:
    $ 40.12万
  • 项目类别:
    Fellowship Award

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