Isometric embeddings, isoperimetric inequalities and geometric nonlinear PDE

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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.