RUI: Problems in Geometric Analysis and General Relativity

RUI:几何分析和广义相对论中的问题

基本信息

  • 批准号:
    1207844
  • 负责人:
  • 金额:
    $ 14.91万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2012
  • 资助国家:
    美国
  • 起止时间:
    2012-08-01 至 2016-07-31
  • 项目状态:
    已结题

项目摘要

The P.I. will study questions in geometric analysis arising both from physics (via the Einstein equation) as well as from geometry (geometric variational problems and scalar curvature). Initial data sets for the Einstein field equation satisfy a nonlinear elliptic system of equations, the Einstein constraint equations. The study of this system has proven to be interesting and fruitful for geometric analysis, and has shed light on the structure of the space of solutions to the Einstein field equation, which is of interest for physics. In previous joint work with R. Schoen, the P.I. developed deformation and gluing techniques for the constraint operator which have been employed (by the P.I. and others) to construct interesting initial data sets, and have led to a better understanding of the structure of asymptotically flat solutions of the constraint equations. Part of the project involves extending results on asymptotics and gluing, including construction of initial data modeling N-body configurations, to include certain matter models, and analyzing the extent to which some of the results that have been developed in the asymptotically flat setting can extend to the asymptotically hyperbolic case. The P.I. will also address several questions on the structure of small-data solutions of the constraints, and as well as on applications of localized scalar curvature deformation. In a second part of the project, the P.I. will continue the study of a variational problem in Kahler and symplectic geometry, the Hamiltonian stationary Lagrangian problem, building on recent joint work with A. Butscher. Another interesting variational problem, the isoperimetric problem (minimizing the area required to enclose a given volume), leads to questions on manifolds-with-density and on spaces relevant to general relativity amenable to research with undergraduates. Solutions to the Einstein field equations are used to model gravitational radiation, strong field phenomena like black holes, isolated gravitational systems, and the large-scale structure of the universe. A more detailed understanding of the space of solutions to the Einstein constraint equations would yield a better understanding of these models. For instance, understanding the asymptotic structure of solutions to the constraints can yield information about models of gravitational radiation. Analysis of the constraint system leads to the construction of initial data with interesting properties, such as solutions in which two or more isolated systems are fused together into a connected solution of the constraints. It would be very interesting for the study of gravitational radiation to numerically implement constructions of small initial data with special asymptotics, and then numerically solve the Einstein field equations; similar comments apply to N-body configurations. An important aspect of the project is to introduce undergraduates to the connections between geometry, analysis and physics. Students will undertake research with the PI during the summer (occurring simultaneously with the Lafayette REU Site program, further enriching the summer research environment in the department), and during the academic year students will be involved in course work, independent study, and research with the PI.
私家侦探将研究从物理(通过爱因斯坦方程)以及几何(几何变分问题和标量曲率)产生的几何分析问题。爱因斯坦场方程的初始数据集满足非线性椭圆方程组,即爱因斯坦约束方程。 这个系统的研究已被证明是有趣的和富有成效的几何分析,并揭示了空间的结构的解决方案的爱因斯坦场方程,这是感兴趣的物理。 在以前的工作与R。肖恩那个私家侦探开发的变形和胶合技术的约束运营商已采用(由P.I.和其他),以构建有趣的初始数据集,并导致了更好地理解的结构渐近平坦的约束方程的解决方案。 该项目的一部分涉及扩展结果的渐近性和胶合,包括建设的初始数据建模N体配置,包括某些物质模型,并分析在何种程度上已经开发的结果在渐近平坦的设置可以扩展到渐近双曲的情况。私家侦探还将解决几个问题的结构小数据的解决方案的约束,以及应用程序的局部标量曲率变形。 在该项目的第二部分,PI。将继续研究卡勒和辛几何中的变分问题,哈密顿定常拉格朗日问题,建立在最近与A。布彻 另一个有趣的变分问题,等周问题(最小化封闭给定体积所需的面积),导致了关于密度流形和与广义相对论相关的空间的问题,适合与本科生一起研究。 爱因斯坦场方程的解被用来模拟引力辐射、黑洞等强场现象、孤立的引力系统和宇宙的大尺度结构。 对爱因斯坦约束方程的解的空间的更详细的理解将产生对这些模型的更好的理解。 例如,理解约束解的渐近结构可以产生关于引力辐射模型的信息。 约束系统的分析导致构造具有有趣性质的初始数据,例如将两个或多个孤立系统融合在一起成为约束的连接解的解。 对于引力辐射的研究来说,数值实现具有特殊渐近性的小初始数据的构造,然后数值求解爱因斯坦场方程是非常有趣的;类似的评论适用于N体构型。 该项目的一个重要方面是向本科生介绍几何,分析和物理之间的联系。学生将在夏季与PI进行研究(与拉斐特REU网站计划同时发生,进一步丰富了部门的夏季研究环境),在学年期间,学生将参与课程工作,独立学习和PI研究。

项目成果

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Justin Corvino其他文献

Multi-localized time-symmetric initial data for the Einstein vacuum equations
爱因斯坦真空方程的多局域时间对称初始数据
A short note on the Bartnik mass
关于巴特尼克弥撒的简短说明
Initial data for the relativistic gravitational N-body problem
相对论引力 N 体问题的初始数据
  • DOI:
    10.1088/0264-9381/27/22/222002
  • 发表时间:
    2010
  • 期刊:
  • 影响因子:
    0
  • 作者:
    P. Chruściel;Justin Corvino;J. Isenberg
  • 通讯作者:
    J. Isenberg
A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres
  • DOI:
    10.1090/s0002-9939-05-07926-8
  • 发表时间:
    2005-06
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Justin Corvino
  • 通讯作者:
    Justin Corvino
On isoperimetric surfaces in general relativity, II
在广义相对论的等周面上,II
  • DOI:
  • 发表时间:
    2009
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Farhan Abedin;Justin Corvino;Shelvean Kapita;Haotian Wu
  • 通讯作者:
    Haotian Wu

Justin Corvino的其他文献

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{{ truncateString('Justin Corvino', 18)}}的其他基金

Between Geometry and Relativity
几何与相对论之间
  • 批准号:
    1740888
  • 财政年份:
    2017
  • 资助金额:
    $ 14.91万
  • 项目类别:
    Standard Grant
Ninety-Nine Years of General Relativity: ESI-EMS-IAMP Summer School on Global Aspects of Mathematical Relativity
广义相对论九十九年:ESI-EMS-IAMP 数学相对论全球方面暑期学校
  • 批准号:
    1406614
  • 财政年份:
    2014
  • 资助金额:
    $ 14.91万
  • 项目类别:
    Standard Grant
RUI: Problems in Geometric Analysis and General Relativity
RUI:几何分析和广义相对论中的问题
  • 批准号:
    0707317
  • 财政年份:
    2007
  • 资助金额:
    $ 14.91万
  • 项目类别:
    Standard Grant
Scalar Curvature and Applications to General Relativity
标量曲率及其在广义相对论中的应用
  • 批准号:
    0071526
  • 财政年份:
    2000
  • 资助金额:
    $ 14.91万
  • 项目类别:
    Fellowship Award

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