Bi-Halfspace and Convex Hull Theorems for Translating Solitons

Abstract

While it is well known from examples that no interesting "halfspace theorem" holds for properly immersed n-dimensional selftranslating mean curvature flow solitons in Euclidean space R n+1 , we show that they must all obey a general "bi-halfspace theorem" (aka "wedge theorem"): Two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori-Yau maximum principle.As an application we classify the convex hulls of all properly immersed (possibly with compact boundary) n-dimensional mean curvature flow self-translating solitons Σ n in R n+1 , up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman-Meeks in 1989, for minimal submanifolds: All of R n , halfspaces, slabs, hyperplanes and convex compacts in R n .

Simply connected translating solitons contained in slabs

“…The curvature estimate is a consequence of an estimate by Schoen and Simon [38]. In Section 2, which is the longest section of this work, we prove Theorem 10, which is a re nement of results contained in the paper by Møller and the author [6]. The proofs are based on a combination of an Omori-Yau maximum principle and barrier arguments.…”

“…Remark 9. From [6] (see also the more general case of ancient ows [7]) it is known that Conv(π(Σ)) is either a line, a strip, a half-plane or the whole R . Therefore π − (∂ Conv(π(Σ))) can be, respectively, only one of the following (i) a vertical plane, (ii) two parallel vertical planes, (iii) the empty set.…”

“…This will nish the proof of Step (i), because of (10). We will do this by using the Omori-Yau maximum principle for properly immersed translaters and we refer to [6] and to [42] for details. Let us assume for contradiction that there exists s * ∈ R such that |s * | > ϱ * and such that sup…”

“…is smooth. From standard computations (see [6]) and using equation 1, one can easily see that on Σ ∩ π − (A s * )…”

Invariant Translators of the Heisenberg Group

“…A general classification is far from being understood. There are many non-graphical examples (those with helicoidal symmetries [12] including some not embedded ones and some others with non-trivial topology [9,20,25]) and some rigidity results [7,14,18,21]. The literature about this topic is quite large and these lists are far from being complete.…”