Isometric embeddings, isoperimetric inequalities and geometric nonlinear PDE

等距嵌入、等周不等式和几何非线性 PDE

• 批准号: RGPIN-2018-04443
• 负责人: Guan, Pengfei
• 金额: $ 4.15万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=690708
• 关键词: Isometric; embeddings; isoperimetric; inequalities; geometric

The proposed research program is centred on fundamental problems in differential geometry and nonlinear PDE: the isometric embedding problem, the isoperimetric type inequalities on general manifolds, and regularity of solutions to nonlinear geometric partial differential equations. ****** The first topic is isometric embedding problem for compact surfaces to three dimensional Riemannian manifold with horizons. When the ambient space is Euclidean space, it is the classical Weyl problem. It is of importance in geometry to consider general ambient space, this also is related to the notions of quasi local masses in general relativity. The most interesting case is that when the ambient space is a anti de Sitter-Schwarzchilds space. ******The second topic concerns various global geometric quantities on manifolds, like volume, surface area, quermassintegrals etc. We would like to establish optimal isoperimetric type inequalities for these geometric quantities. Our approach will be based on nonlinear partial differential equations of parabolic type. For each pair of geometric quantities, we would like to design a curvature flow such that: along the flow, one quantity is preserved and another is monotone. The key is to prove the longtime existence and convergence of the flow.****** The last topic addresses some longstanding regularity problems of curvature type equations. Pogorelov type counter-examples indicate that interior regularity fails for Monge-Amp\`ere equation when dimension is larger or equal to three. One longstanding open problem is that, if interior estimate holds for scalar curvature equation and $\sigma_2$ Hessian equation. These geometric equations are of fundamental importance, for example, scalar curvature equation naturally arising from the isometric embedding problems. A breakthrough will have great impact in geometric analysis.****** A common thread linking our program is the analysis of the geometric fully nonlinear equations. These equations are the main subjects of the research program. Besides the regularity and existence of solutions of these equations (which are still important subjects of the study), there emerge some new directions of research from the proposed problems. One main challenge is for the isometric embedding problem discussed is the existence of homotopic paths, we propose a novel approach using geometric flows in combination with elliptic method. The flow approach will also be devised to establish isoperimetric type inequalities: explore the variational properties of the associated functionals to design a flow with appropriate monotonicity properties. For the regularity problems of solutions to geometric nonlinear PDE, we propose new ideas to deal with the issue. ****** Our objective is to develop various analytic tools for geometric nonlinear partial differential equations, investigate structures of solutions and derive geometric consequences.

拟议的研究计划集中在差异几何和非线性PDE中的基本问题上：等距嵌入问题，一般流形的等二仪类型不平等以及非线性几何学偏微分微分方程的解决方案的规律性。 ******第一个主题是将紧凑型表面的等距嵌入问题与三维riemannian歧管的层次。当环境空间是欧几里得空间时，它是经典的韦里问题。在几何学中，考虑一般的环境空间很重要，这也与一般相对性中的准局部质量概念有关。最有趣的案例是，当环境空间是一个反de Sitter-Schwarzilds空间时。 ******第二个主题涉及多种流形的各种全局几何量，例如体积，表面积，QuermassIntegrals等。我们想为这些几何数量建立最佳的等速型不平等。我们的方法将基于抛物线类型的非线性偏微分方程。对于每对几何量，我们想设计一个曲率流，使得：沿流动，保留一个数量，另一个是单调的。关键是要证明流量的长期存在和收敛。******最后一个主题解决了曲率类型方程的一些长期规律性问题。 pogorelov类型的反例表明，当尺寸较大或等于三个时，Monge-Amp \'方程的内部规律性失败。一个长期的开放问题是，如果室内估计为标态曲率方程和$ \ sigma_2 $ hessian方程。这些几何方程至关重要，例如，标量曲率方程是由等距嵌入问题自然引起的。突破将在几何分析中产生重大影响。这些方程是研究计划的主要主题。除了这些方程式的规律性和解决方案（仍然是研究的重要主题）外，还从提出的问题中出现了一些新的研究方向。一个主要的挑战是讨论的等距嵌入问题是存在同型路径，我们提出了一种新的方法，使用几何流与椭圆方法结合使用。还将设计流程图以建立等等类型的不等式：探索相关功能的变异性质，以设计具有适当单调性能的流动。对于几何非线性PDE解决方案的规律性问题，我们提出了解决这个问题的新想法。 *****我们的目标是开发各种分析工具，用于几何非线性偏微分方程，研究解决方案的结构并得出几何后果。