UC Santa Cruz

The equidistant surfaces in a Fuchsian manifold form a canonical foliation by constant mean curvature surfaces. It is therefore natural to ask whether a prescribed leaf of this foliation can be reached by evolving an arbitrary graphical surface. We answer this question affirmatively using the modified mean curvature flow. Let (M³, ḡ)=(ℝ×Σ, dr²+cosh²(r)g₀) be a Fuchsian manifold, where (Σ,g₀) is a closed hyperbolic surface. For any σ∈(−2, 2), we consider the flow ??/??=−(?−?)? starting from an arbitrary smooth closed geodesic graph over the central totally geodesic surface. We prove that the flow exists smoothly for all time, remains graphical, and converges smoothly to the unique equidistant surface Σ(?_?), where ?_?=arctanh(?/2) whose mean curvature is ?. The initial graph is required only to be strictly graphical; no smallness assumption or quantitative angle–height pinching condition is imposed. The main difficulty is that strict graphicality alone does not provide a uniform bound for the reciprocal of the angle function. We first derive a time-dependent gradient estimate, sufficient to preserve graphicality and exclude finite-time singularities. We then obtain polynomial curvature control and use a blow-up argument to establish a time-independent gradient bound. The resulting uniform curvature estimates, together with the barrier structure supplied by the equidistant foliation, lead to smooth convergence to Σ(?_?).