Isometric embeddings, isoperimetric inequalities and geometric nonlinear PDE

所提出的研究计划集中在微分几何和非线性偏微分方程中的基本问题：等距嵌入问题，一般流形上的等周型不等式，以及非线性几何偏微分方程解的正则性。*第一个主题是紧致曲面到三维水平黎曼流形的等距嵌入问题。当环境空间是欧氏空间时，这是经典的Weyl问题。考虑一般环境空间在几何学中具有重要意义，这也与广义相对论中的准局部质量概念有关。最有趣的情况是，当周围空间是反de Sitter-Schwarzchads空间时。*第二个主题涉及流形上的各种整体几何量，如体积、表面积、积分等。我们想要建立这些几何量的最优等周型不等式。我们的方法将基于抛物型的非线性偏微分方程组。对于每一对几何量，我们想要设计一个曲率流：沿着流，一个量是保留的，另一个是单调的。关键是证明流的长期存在和收敛。*最后一个主题讨论了一些长期存在的曲率型方程的正则性问题。Pogorelov型反例表明，当维度大于或等于3时，Monge-Amp方程的内部正则性失效。一个长期悬而未决的问题是，如果标量曲率方程和$\sigma_2$Hessian方程的内部估计成立。这些几何方程是非常重要的，例如，由等距嵌入问题自然产生的标量曲率方程。一项突破将对几何分析产生重大影响。*连接我们的程序的一个共同线索是几何完全非线性方程的分析。这些方程是研究计划的主要主题。除了这些方程的正则性和解的存在性(这些仍然是研究的重要主题)之外，从提出的问题中还出现了一些新的研究方向。针对等距嵌入问题的一个主要挑战是同伦路径的存在性，我们提出了一种结合几何流和椭圆方法的新方法。流方法也将被设计来建立等周型不等式：探索相关泛函的变分性质以设计具有适当单调性的流。对于几何非线性偏微分方程解的正则性问题，我们提出了新的解决思路。*我们的目标是开发各种几何非线性偏微分方程解的分析工具，研究解的结构，并推导几何推论。

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.