Gradient Estimates For Closed Starshaped Hypersurface With Prescribed Curvature

In this paper, we mainly study the gradient estimate for closed star-shaped hypersurface with prescribed curvature and curvature measure.There are some degree theories and continuation methods when studying the existence of solution. A priori estimate is necessary,and gradient estimate is a part of a priori estimate.The paper includes precribed mean curvature case and precribed Gauss curvature measure case .It is organized as follows:1. 2-dimentional gradient estimate for hypersurface with precribed mean curvature, recalling the Treibergs and Wei's theorem in 1983.2 . 2-dimentional and n-dimentional gradient estimate for hypersurface with precribed Gauss curvature measure,i.e.theorem 2.3.1 which is main result of this paper.