Translating Solutions Of Mean Curvature Flow And H~k Flow

This thesis investigates the global properties and the existence of translating so-lutions to H~k(k > 0) flow on unbounded domains where H~k flow is mean curvatureflow if k = 1 and the asymptotic behavior at infinity of translating solutions to meancurvature flow with a constant forcing term. We will use analytic approach to studythese geometrical partial equations.Firstly, we systematically investigate the properties of radially symmetric trans-lating solutions of H~k flow in Euclidean space, and obtain the optimal estimates up tothe second derivative of solutions and the precise asymptotic expansion of solutions .Then, we prove interior gradient estimate for translating solutions of H~k flow, and gen-eralize the corresponding results about the equations of minimal surface type whichhave been established by famous mathematicians. Moreover, we prove the Liouvilletype theorem by using gradient estimate. The conditions of Liouville theorem are op-timal if k is small. More importantly, we study the existence of smooth translatingsolutions for H~k flow on bounded mean convex domain and unbounded mean convexdomain like a cone or a U-type domain, respectively.Secondly, we extend some of the previous methods and techniques to the study ofasymptotic behavior of rotationally symmetric translating solutions to mean curvatureflow with a constant forcing term in Minkowski space, discuss two cases that the hyper-surface translates upward and downward, respectively, and obtain different asymptoticexpansions. Besides, we also obtain the optimal derivative estimates of solutions.The main theorems in this thesis are theorems 3.1-3.3, 3.8-3.10 and theorems 4.1-4.2.