A characterization of the grim reaper cylinder

Abstract

Abstract. In this article we prove that a connected and properly embedded translating soliton in R 3 with uniformly bounded genus on compact sets which is C 1 -asymptotic to two planes outside a cylinder, either is flat or coincides with the grim reaper cylinder.

Equilibrium of Surfaces in a Vertical Force Field

“…It is interesting to mention • The case of ϕ(z) = z: it corresponds with translating solitons, that is, surfaces in R 3 such that t → Σ + t e 3 is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point: H = e ⊥ 3 . Recent advances in the understanding of its local and global geometry can be found in [4,8,[12][13][14]16,17,23,25] • The case of ϕ(z) = α log z, α=const. It includes the two dimensional examples analogues of the catenaries (when α = 1).…”

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.…”

Uniqueness of the $[φ,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour

“…Some particular cases of these surfaces have been the key in the development of some issues of the differential geometry theory. We highlight the works [4,5,9,14,15] for Translating soliton for the mean curvature flow, when ϕ is the identity and the works [2,3,8] for singular α-minimal surfaces, when ϕ(x 3 ) = αlog(x 3 ) for α = 0. In fact, the special case α = 1, from a physically point of view [13], represents a membrane with intrinsic force vanishes under a gravitational field.…”

“…Roughly, if we have a surface Σ that looks like a [ϕ, e 3 ]-catenary cylinder G h at infinity, in the sense of the definition 2.2 in Section 2, then our surface coincides with some [ϕ, e 3 ]-catenary cylinder with the same behaviour G h . Motivated by the work [9] of F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk for the grim reaper cylinder translating soliton, we increase the family of [ϕ, e 3 ]-minimal surfaces where these types of results hold under different assumptions of asymptotic behaviour when ϕ is a strictly increasing convex diffeomorphism such that e −ϕ is integrable. From [11,12], this family of functions ϕ is the natural candidate to consider for generalizing the result of [9].…”

“…Motivated by the work [9] of F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk for the grim reaper cylinder translating soliton, we increase the family of [ϕ, e 3 ]-minimal surfaces where these types of results hold under different assumptions of asymptotic behaviour when ϕ is a strictly increasing convex diffeomorphism such that e −ϕ is integrable. From [11,12], this family of functions ϕ is the natural candidate to consider for generalizing the result of [9]. The statement of the main result of this paper tells us the following We would like to point out that we generalize this uniqueness for [ϕ, e 3 ]minimal surface assuming that Σ is a vertical graph.…”