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In the first part of this paper we obtain a new gradient estimate for starshaped hypersurfaces moving in the direction of their normals with speed a function of the principal curvatures. For convex hypersurfaces moving by their curvature normal vectors, a uniform gradient estimate for the support function was obtained by Chow-Gulliver [CGI]. The proof was based on a fixed plane version of the Aleksandrov reflection method. Furthermore, in Chow-Gulliver [CG2], the Aleksandrov reflection method was applied to embedded hypersurfaces moving by their curvature vectors. In section 2, we apply their reflection result to obtain a uniform gradient estimate for the radial function of a starshaped hypersurface moving by its curvature vector, which holds outside a certain compact set depending only on the initial hypersurface. This gradient estimate should be useful in proving long-time existence theorems for starshaped hypersurfaces expanding by nonhomogeneous functions of curvature. For homogeneous of degree one functions, long-time existence theorems were proved by Gerhardt [G] and Urbas [U2]. We also give a new geometric proof of the gradient estimate for the support function for evolving convex hypersurfaces in [CGI]. The gradient estimate for the support function was used as a first step in proving long-time existence by Chow-Tsai [CT1] in the case of expanding convex plane curves, Chow-Tsai [CT2] to remove the homogeneity assumption on the speed of the flow in the theorems of Gerhardt [G], Huisken [H] and Urbas [Ul] for expanding convex hypersurfaces, and Chow-Liou-Tsai [CLT] to the equation ut — F(Au + nu) on 5TM, where F is an arbitrary increasing function. Finally, see Tsai [T] for the global existence and convergence of expanding flows of starshaped plane curves with arbitrary nonhomogeneous speeds. Tsai proved a uniform gradient estimate for the radial function via intricate maximum principle arguments. Using different methods we partially gener-

Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations