Simply connected translating solitons contained in slabs

Francesco Chini

doi:10.1515/geofl-2020-0101

Abstract

AbstractIn this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.

Highlights

In this work we show that -dimensional, connected, translating solitons of the mean curvature ow embedded in a slab of R with entropy strictly less than must be mean convex and thanks to a result by Spruck and Xiao are convex
Such -dimensional convex translating solitons have been completely classi ed, up to an ambient isometry, as vertical planes, grim reaper cylinders, ∆-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater
New examples by Ho man, Martín and White show that the bound on the entropy is necessary

Summary

Introduction

In [1], Brendle proved that any properly embedded -dimensional mean curvature self-shrinker in R which is homeomorphic to an open subset of the sphere must be a round sphere, or a cylinder or a plane, solving two problems posed by Ilmanen (see 14 and 15 in [27]). Let Σ ⊆ R be a complete, embedded, translater satisfying the following assumptions: (i) Σ is connected, (ii) λ(Σ) < , He proved that a translater Σ ⊆ R satisfying the following entropy bound λ(Σ) ≤ λ(S × R) = π ≈ .

Results

Conclusion