Paper · Jul 29, 2021

Uniqueness of the $[φ, e_{3}]$ -catenary cylinders by their asymptotic behaviour

A. L. Martínez-Triviño, J. P. dos Santos

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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 $C^{\infty}$ -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.

Mean curvature flow Self-shrinkers & translators Stability & uniqueness

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Uniqueness of the $[φ, e_{3}]$ -catenary cylinders by their asymptotic behaviour

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Uniqueness of the [φ,e3​]-catenary cylinders by their asymptotic behaviour

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Uniqueness of the $[φ, e_{3}]$ -catenary cylinders by their asymptotic behaviour