Some Curvature Problems On Riemannian Manifolds

Fully nonlinear elliptic and parabolic partial differential equations, as well as their applications in geometry have received extensive investigation. In this thesis we focus on the application in curvature problems on Riemannian manifolds. We mainly study three problems, namely hypersurfaces with prescribed Weingarten curvature, complete conformal metric with prescribed curvature, and surfaces ex-panding by the power of Gauss curvature.Let N be a (n+l)-dimensional Riemannian manifold, and M. be a n-dimensional submanifold. Given a smooth function Ψ in N, we study the existence of an em-bedding φ：Mâ†'N such that f(κ)=Ψ, where κ=(κ1,…,κn) are principal curvatures of hypersurface φ>(M), and f is a symmetric function. The prescribing curvature problems have been studied by many authors. There is a vast body of works dealing with the case N=Rn+1. We study the case when N is a Riemannian manifold. Assuming that N admits a normal Gaussian coordinate system, we obtained, under some proper conditions, the existence of closed hypersurfaces with prescribed Weingarten for a large class of curvature functions f. Our geometric model constitutes a large class of Riemannian manifolds which include space forms, and more generally, warped product manifolds.Let M be a compact n-dimensional Riemannian manifold with boundary (?)M,, and denote the metric tensor, Ricci tensor and Scalar curvature by g, Ricg and Rg. Gursky and Viaclovsky introduced a modified Schouten tensor AT, of the form Ric+cTRg, where cT is a constance depending on the parameter T. This curvature arises naturally in decomposition of Riemannian curvature tensor. We consider the problem of deforming the metric in the conformal class [g] to certain complete metric g-such that f(κ)=ψ(?) χ∈M=M\(?)M, where ψ is a smooth function in M, and κ=(κ1,…,κn) are the eigenvalues of Agt. This is equivalent to solving a fully nonlinear elliptic equation with infinite boundary condition. Assuming T> n-1we prove that this problem is solvable for a large class of curvature functions f. In particular, we show every smooth compact n-dimensional manifold with boundary admits a complete Riemannian metric g, whose Ricci curvature and Scalar curvature satisfy det(Ric-Rg)=const.>0.In the last few decades, many geometers utilised parabolic partial differential equations to understand the geometry and the topology of manifolds, which create an exciting and fruitful area, namely the geometric flow. For example, Hamilton introduced the Ricci flow, by which he aimed to study the geometrization conjecture of Thurston. Following the works of Hamilton, this conjecture was finally proved by Perelman. This is a major achievement in this area. In another aspect, Andrews proved a conjecture of Firey that convex surfaces evolving by their Gauss curvature become spherical in shape as it shrinks. This flow was introduced by Firey as a model of the rolling stones. As a natural generalisation, we consider the flow of surfaces in R3for some negative power of the Gauss curvature K. We show that strictly convex surfaces expanding with normal velocity K-α,1/2<α≤1, converge to infinity spherically in finite time. This result closes a gap in the powers considered by previous authors, that is, for α∈(0,1-2] by Urbas and Huisken and for α=1by Schniirer.