Uniqueness of the [formula omitted]-catenary cylinders by their asymptotic behaviour

Abstract

We establish a uniqueness result for the [φ,e→3]-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk proved that, if Σ is a properly embedded translating soliton with locally bounded genus and C1-asymptotic to two vertical planes, outside a cylinder, then Σ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of [φ,e→3]-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.