本文選題：完全非線性Hessian方程 + 黎曼流形�。� 參考：《哈爾濱工業(yè)大學(xué)》2016年博士論文

【摘要】：微分幾何中的許多問(wèn)題都可以轉(zhuǎn)化為解一個(gè)完全非線性Hessian方程。例如Calabi猜想等價(jià)于求解緊Kahler流形上的一個(gè)復(fù)Monge-Ampere方程。而Minkowski問(wèn)題等價(jià)于求解球面上的一個(gè)Monge-Ampere方程。共形幾何中的k-Yamabe問(wèn)題等價(jià)于求解閉流形上的完全非線性橢圓型Hessian方程。通過(guò)構(gòu)造k-Yamabe流,這一問(wèn)題還可以轉(zhuǎn)化為求解完全非線性拋物型Hessian方程。另外,嚴(yán)格凸超曲面在其高斯曲率作用下的形變,利用高斯映射能夠化為一個(gè)拋物型的Monge-Ampere方程。可見,完全非線性Hessian方程與幾何等領(lǐng)域的研究密切相關(guān)。因此,對(duì)完全非線性Hessian方程的研究具有十分重要的理論意義和應(yīng)用價(jià)值。本文研究了幾類帶邊黎曼流形上的完全非線性Hessian方程,證明了其容許解的先驗(yàn)C2估計(jì)。由Evans-Krylov定理及Schauder理論可以得到解的更高階估計(jì)。從而,應(yīng)用連續(xù)性方法和拓?fù)涠壤碚摰玫搅朔匠探獾拇嬖谛�。具體地,本文得到以下成果：首先,對(duì)一類完全非線性橢圓型Hessian方程的Dirichlet問(wèn)題證明了光滑解的存在性。作為這一結(jié)果的推論,共形幾何中的一類預(yù)定負(fù)曲率問(wèn)題有解,即在帶邊黎曼流形(Mg)上存在共形度量g,使得在流形的邊界(?)M上g和g相同,并且在新的度量下參數(shù)化的(modified) Schouten張量Agt滿足給定的方程。其次,對(duì)MT:= M x (0,T]上一類完全非線性拋物型Hessian方程的第一初邊值問(wèn)題證明了光滑解的存在性。與橢圓的情形一樣,通過(guò)證明解的先驗(yàn)C2估計(jì),得到了該問(wèn)題的光滑解。為了避免對(duì)流形的邊界添加過(guò)多的幾何假設(shè),這里利用了下解的存在性。完全非線性算子只需滿足結(jié)構(gòu)性條件。只有在證明梯度估計(jì)時(shí),用到了一條技術(shù)性假設(shè)。最后,對(duì)一類完全非線性Hessian方程的障礙問(wèn)題證明了C1,1解的存在性。這類問(wèn)題常出現(xiàn)在尋找具有給定曲率限制的最大(或最小)超曲面的問(wèn)題當(dāng)中。作為應(yīng)用,用同樣的方法證明了共形幾何中一類預(yù)定負(fù)曲率方程的障礙問(wèn)題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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O186.12;O175

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本文編號(hào)：1919197