On the topology of translating solitons of the mean curvature flow

Abstract

Abstract. In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.

A characterization of the grim reaper cylinder

“…Then, by applying the strong maximum principle to ξ 2 H −1 , we deduce that ξ 2 is identically zero. This implies that the Gauß curvature of M is zero and so M must coincide with a grim reaper cylinder (see [MSHS15,Theorem B]). …”

Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

“…Moreover the flow parameter is given by where ∇ P and div P are, respectively, the Riemannian connection and divergence in (P, g 0 ). This notion of translating soliton has been extensively studied in Euclidean spaces, see for instance [29], [4], [14], [38], [39], [41], [45], [48], [49] only to quote a few examples of the vast literature on the subject. Our definition is the natural setting to these special flows in Riemannian products I × P with I ⊂ R.…”

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

“…The paper [CE16] also showed a halfspace theorem (by using the half-catenoid-like "self-shrinking trumpets" from [KM14] as barriers) and [IPR18] showed a "Frankel property" for self-shrinkers (meaning: when it so happens that all minimal surfaces in a space must intersect, as in [Fr66] and [PW03]). Additionally, for selftranslaters, a few significant geometric classification and nonexistence results are now known, see [Wa11], [Sh11], [MSS14], [Mø14], [Ha15], [Pé16], [IR17], [Bu18] and [HIMW18-1], but these do not directly address the question of (bi-)halfspace and convex hull properties.…”

“…In [Sh11] and [Sh15], Shahriyari proved that there are no complete 2dimensional translaters which are graphical over a bounded domain. This fact was later generalized by Møller in [Mø14] (see [MSS14] for the halfcylinder case), where he proved that there are no properly embedded without boundary n-dimensional self-translaters contained in a cylinder of the kind Ω × R, where Ω ⊆ R n is bounded:…”