Abstract
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.
Highlights
The purpose of this survey is to illustrate some recent rigidity results, obtained by the authors, about the prescribed mean curvature problem on a complete Riemannian manifold
Do Carmo and Lawson in [47] investigated geodesic graphs in H +1 with constant mean curvature ∈ [1, 1], both over horospheres and hyperspheres, and proved the following Bernstein type theorem: if Σ is over a horosphere, Σ is a horosphere if Σ is over a hypersphere, Σ is a hypersphere
There exists no entire graph in the Schwarzschild and ADSSchwarzschild space, over a complete, that is a soliton with respect to the field √ ( )
Summary
The purpose of this survey is to illustrate some recent rigidity results, obtained by the authors, about the prescribed mean curvature problem on a complete Riemannian manifold. Do Carmo and Lawson in [47] investigated geodesic graphs in H +1 with constant mean curvature ∈ [1, 1], both over horospheres and hyperspheres, and proved the following Bernstein type theorem: if Σ is over a horosphere, Σ is a horosphere (in particular, = ±1) if Σ is over a hypersphere, Σ is a hypersphere. Their result is part of a more general statement, that draws the rigidity of a properly embedded CMC hypersurface Σ from the rigidity of its trace on the boundary at infinity ∞H +1. Existence for the prescribed mean curvature equation on more general products ×h R is studied in [17]
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