Translating solitions of the mean curvature flow

Abstract

In the first chapter of this thesis, after a brief introduction to the mean curvature ow and translating solitons, we present the classic examples of the latter ones. It is well known that translating solitons are related to minimal surfaces [Ilm94]. Obviously, this relationship is important because it allows to use classical results of the theory of minimal surfaces to study translating solitons. In this spirit, the maximum principle, stated as its geometric counterpart, the tangency principle, is the main tool of the second chapter of the thesis, which begins with the proof of the results of non-existence of translating solitons. We prove that there are no non-compact translating solitons contained in a solid cylinder (Theorem 2.1.2). We also rule out the existence of certain compact embedded translating solitons with two boundary components (Theorem 2.1.5). Then, by comparison with a tilted grim reaper cylinder, we obtain an estimate of the maximum height that a compact translating soliton embededd in R3 can achieve; this estimate is in terms of the diameter of the boundary curved of the translator (Theorem 2.2.1). Another application to the tangency principle is to study graphical perturbations of translating solitons, which allows us to easily prove the characterization of the translating paraboloid given in [MSHS15, Theorem A]. On the other hand, we use the method of moving planes to show that a compact embedded translating soliton contained in a slab and with boundary components given by two convex curves in the parallel planes determining the slab inherits all the symmetries of its boundary (Theorem 2.4.1). The main result of the thesis is presented in the third and last chapter and it is a characterization of grim reaper cylinders as properly embedded translators with uniformly bounded genus and asymptotic to two half-planes whose boundaries are contained in the boundary of a solid cylinder with axis perpendicular to the direction of translation (Theorem 3.0.2). The proof is quite elaborated and heavily uses analytic tools developed by Brian White: a compactness theorem for sequences of minimal surfaces properly embedded into three-dimensional manifolds with locally uniformly bounded area and genus, as well as a barrier principle. As mentioned above, the key ingredient to use these results of White is to consider translating solitons as minimal surfaces in the so-called Ilmanen's metric and to establish the good relation between these surfaces in both (usual Euclidean and Ilmanen) metrics, in particular with respect to their asymptotic behavior.