黎曼流形上椭圆型及抛物型完全非线性Hessian方程的研究

发布时间：2018-05-21 12:33

本文选题：完全非线性Hessian方程 + 黎曼流形　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：微分几何中的许多问题都可以转化为解一个完全非线性Hessian方程。例如Calabi猜想等价于求解紧Kahler流形上的一个复Monge-Ampere方程。而Minkowski问题等价于求解球面上的一个Monge-Ampere方程。共形几何中的k-Yamabe问题等价于求解闭流形上的完全非线性椭圆型Hessian方程。通过构造k-Yamabe流,这一问题还可以转化为求解完全非线性抛物型Hessian方程。另外,严格凸超曲面在其高斯曲率作用下的形变,利用高斯映射能够化为一个抛物型的Monge-Ampere方程。可见,完全非线性Hessian方程与几何等领域的研究密切相关。因此,对完全非线性Hessian方程的研究具有十分重要的理论意义和应用价值。本文研究了几类带边黎曼流形上的完全非线性Hessian方程,证明了其容许解的先验C2估计。由Evans-Krylov定理及Schauder理论可以得到解的更高阶估计。从而,应用连续性方法和拓扑度理论得到了方程解的存在性。具体地,本文得到以下成果：首先,对一类完全非线性椭圆型Hessian方程的Dirichlet问题证明了光滑解的存在性。作为这一结果的推论,共形几何中的一类预定负曲率问题有解,即在带边黎曼流形(Mg)上存在共形度量g,使得在流形的边界(?)M上g和g相同,并且在新的度量下参数化的(modified) Schouten张量Agt满足给定的方程。其次,对MT:= M x (0,T]上一类完全非线性抛物型Hessian方程的第一初边值问题证明了光滑解的存在性。与椭圆的情形一样,通过证明解的先验C2估计,得到了该问题的光滑解。为了避免对流形的边界添加过多的几何假设,这里利用了下解的存在性。完全非线性算子只需满足结构性条件。只有在证明梯度估计时,用到了一条技术性假设。最后,对一类完全非线性Hessian方程的障碍问题证明了C1,1解的存在性。这类问题常出现在寻找具有给定曲率限制的最大(或最小)超曲面的问题当中。作为应用,用同样的方法证明了共形几何中一类预定负曲率方程的障碍问题C1,1解的存在性。
[Abstract]:Many problems in differential geometry can be transformed into solutions to a completely nonlinear Hessian equation. For example, Calabi conjecture is equivalent to solving a complex Monge-Ampere equation on a compact Kahler manifold. The Minkowski problem is equivalent to solving a Monge-Ampere equation on a sphere. The k-Yamabe problem in conformal geometry is equivalent to solving completely nonlinear elliptic Hessian equations on closed manifolds. By constructing k-Yamabe flow, the problem can also be transformed into solving completely nonlinear parabolic Hessian equations. In addition, the deformation of strictly convex hypersurface under its Gao Si curvature can be transformed into a parabolic Monge-Ampere equation by Gao Si mapping. It can be seen that the completely nonlinear Hessian equation is closely related to the study of geometry and other fields. Therefore, the study of completely nonlinear Hessian equations is of great theoretical significance and practical value. In this paper, we study some completely nonlinear Hessian equations on Riemannian manifolds with edges, and prove a priori C2 estimate of their admissible solutions. Higher order estimates of solutions can be obtained from Evans-Krylov theorem and Schauder theory. Thus, the existence of the solution of the equation is obtained by using the continuity method and the topological degree theory. In this paper, the following results are obtained: firstly, the existence of smooth solutions is proved for the Dirichlet problem of a class of completely nonlinear elliptic Hessian equations. As a corollary of this result, a class of predetermined negative curvature problems in conformal geometry have solutions, that is, there exists a conformal metric g on the Riemannian manifold with edges, so that g and g are the same on the boundary of the manifold. And the parameterized Schouten Zhang Liang Agt satisfies the given equation under the new metric. Secondly, we prove the existence of smooth solutions for the first initial-boundary value problem of a class of completely nonlinear parabolic Hessian equations on MT: = M x 0 T]. As in the case of an ellipse, the smooth solution of the problem is obtained by proving the prior C2 estimate of the solution. In order to avoid adding too many geometric assumptions to the boundary of convection, the existence of lower solution is used. Completely nonlinear operators only need to satisfy structural conditions. Only when the gradient estimation is proved, a technical assumption is used. Finally, the existence of C _ 1N _ 1 solution for a class of completely nonlinear Hessian equations is proved. Such problems often arise in finding the maximum (or minimum) hypersurfaces with given curvature constraints. As an application, the existence of C _ 1N _ 1 solutions for a class of predetermined negative curvature equations in conformal geometry is proved by the same method.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.12;O175

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