Year: 2025
Type: article
Abstract: 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 † ޒ j x 2 †g over †.The smooth solution u to (1-1) is the height function of the minimal graph M in † .ޒ Therefore, we call u a minimal graphic function on †.When † is Euclidean space ޒ 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 ޒ 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 ޒ n satisfies sublinear growth for its negative part, i.e., lim supthen u is a constant.The condition (1-2) is sharp since any affine function is a minimal graphic function on ޒ n .When the minimal graphic function u on ޒ n has the uniformly bounded gradient, Moser [1961] proved u is affineThe author is supported by NSFC 12371053.
Source: Analysis & PDE
Author: Qi Ding
Institution: Fudan University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Award: G3221272539
OpenAlex ID: W4416013536