Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature

Let be a complete Riemannian manifold of nonnegative Ricci curvature.We prove a Liouville-type theorem: every smooth solution u to the minimal hypersurface equation on is a constant provided u has sublinear growth for its negative part.Here, the sublinear growth condition is sharp.Our proof relies on a gradient estimate for minimal graphs over with small linear growth of the negative parts of graphic functions via iteration.which is a nonlinear partial differential equation describing the minimal graph M D f.x; u.x// 2 j x 2 g over .The smooth solution u to (1-1) is the height function of the minimal graph M in . Therefore, we call u a minimal graphic function on .When is Euclidean space n , equation (1-1) has been studied successfully by many mathematicians.Bombieri, De Giorgi, and Miranda [Bombieri et al. 1969b] (see also [Gilbarg and Trudinger 1983]) proved interior gradient estimates for solutions to the minimal hypersurface equation on n , where the 2-dimensional case had already been obtained in [Finn 1954].As a corollary, they immediately got a Liouville-type theorem in [Bombieri et al. 1969b] as follows.Theorem 1.1.If a minimal graphic function u on n satisfies sublinear growth for its negative part, i.e., lim supx!1 maxf u.x/; 0g jxj D 0;(1-2) then u is a constant.The condition (1-2) is sharp since any affine function is a minimal graphic function on n .When the minimal graphic function u on n has the uniformly bounded gradient, Moser [1961] proved u is affine