Non-Parametric Mean Curvature Flow with Prescribed Contact Angle in Riemannian Products

Jean-Baptiste Casteras,Jorge H De Lira,Esko Heinonen,Ilkka Holopainen

doi:10.1515/agms-2020-0132

Abstract

Abstract Assuming that there exists a translating soliton u ∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to u ∞ + Ct as t →∞. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of Ω and Ricci curvature in Ω.

Highlights

We study a non-parametric mean curvature ow in a Riemannian product N × R represented by graphsMt := x, u(x, t) : x ∈ Ω (1.1)with prescribed contact angle with the cylinder ∂Ω × R
Assuming that there exists a translating soliton u∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature ow with the same prescribed contact angle converges to u∞ + Ct as t → ∞
The boundary condition above can be written as ν, γ = φ, (1.3)

Summary

Introduction

We study a non-parametric mean curvature ow in a Riemannian product N × R represented by graphs. Zhou [8] studied mean curvature type ows in a Riemannian product M × R and proved the longtime existence of the solution for relatively compact smooth domains Ω ⊂ M He extended the convergence result of Altschuler and Wu to the case M is a Riemannian surface with nonnegative curvature and Ω ⊂ M is a smooth bounded strictly convex domain; see [8, Theorem 1.4]. We circumvent this obstacle by modifying the method of Korevaar [6], Guan [4] and Zhou [8] and obtain a uniform gradient estimate in an arbitrary relatively compact smooth domain Ω ⊂ N provided there exists a translating soliton with speed C and with the prescribed contact angle condition (1.3) Towards this end, let d be a smooth bounded function de ned in some neighborhood of Ωsuch that d(x) = miny∈∂Ω dist(x, y), the distance to the boundary ∂Ω, for points x ∈ Ω su ciently close to ∂Ω.

We can compute Lh as

Hence we obtain

Hence we obtain an upper bound for the radius