Translating solitons of the mean curvature flow

Abstract

The evolution of geometric objects with respect to a time variable (t) is a field of intense study in the framework of Riemannian Geometry. A good example is the work of Perelman which led to the proof of the geometrization conjecture of Thurston and, consequently, to the Poincaré conjecture. In general, two types of geometric objects are distinguished: the intrinsic ones, such as the metric of a manifold, and the extrinsic ones, such as an embedding (or immersion) of a manifold into another one. These two types of objects give rise to two types of flows: the intrinsic ones, such as the Ricci flow, and the extrinsic ones, such as the mean curvature flow. In these particular flows the object that evolves is precisely the metric and the immersion respectively. The mean curvature flow is maybe the most important geometric evolution equation of submanifolds in Riemannian manifolds. Intuitively, a family of smooth submanifolds evolves under mean curvature flow if the velocity at each point of the submanifold is given by the mean curvature vector at this point, that is, the submanifolds move at each point in the direction of the corresponding normal unit vector and with speed equals to its scalar mean curvature. For example, a sphere in the Euclidean space evolves under mean curvature flow by shrinking inward until it collapses in a finite time to a point. In the literature there are different approaches to the mean curvature flow: from the geometric measure theory, from level set flows and from partial differential equations. We will follow the latter approach, which began with the pioneer work of Huisken “Flow by mean curvature of convex surfaces into spheres” published in 1984. The evolution equation can develop singularities, that is, solutions may become non-smooth in finite time. For example, it is known that closed submanifolds (i.e., compact submanifolds without boundary) of the Euclidean space remain smooth for a finite time during their evolution under mean curvature flow and then they develop a singularity. Thus, a particularly interesting topic in this theory is to study the behaviour of singularities. In order to do that two types of singularities are often distinguished: type I singularities, which are those cases with a better control of the growth of the norm of the second fundamental form, and type II singularities, which are all the other singularities. As the definition suggests, type II singularities are more difficult to treat than type I singularities. In the case of hypersurfaces, Huisken and Sinestrari proved that if the initial surface is mean convex (that is, if the scalar mean curvature is nonnegative) and it develops a type II singularity, then the limit surface obtained with a certain technique of rescaling of the flow is convex and satisfies the equation H=v⊥, where H is the mean curvature vector, v is a constant vector and the superscript ⊥ denotes projection onto the normal bundle. A surface whose mean curvature vector satisfies the above equation is called translating soliton of the mean curvature flow. Geometrically it evolves moving in the direction of v with speed |v|, that is, with fixed direction and constant speed given by the vector v. Thus, it does not change its shape during the evolution, it simply translates. Therefore, translating solitons are eternal solutions of the flow, that is, their evolution exists for all times. Translating solitons arise not only in the study of singularities, but also in the general investigation of the mean curvature flow. For example, again in the case of hypersurfaces in the Euclidean space, Hamilton proved that any strictly convex eternal solution to the mean curvature flow where the mean curvature assumes its maximum value at a point in space-time must be a translation soliton. Moreover, translating solitons are interesting examples of the mean curvature flow because they are precise solutions in the sense that their evolution is known, which is very hard to determine in general. In the first chapter of this thesis, after a brief introduction to the mean curvature flow and translating solitons, we present the classic examples of the latter ones. It is well known that translating solitons are related to minimal surfaces. 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. We also rule out the existence of certain compact embedded translating solitons with two boundary components. Then, by comparison with a tilted grim reaper cylinder, we obtain an estimate of the maximum height that a compact translating soliton embededd in the three dimensional Euclidean space can achieve; this estimate is in terms of the diameter of the boundary curve of the translator. Another application to the tangency principle is to study graphical perturbations of translating solitons, which allows us to easily prove a characterization of the translating paraboloid. 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. 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. The proof is quite elaborated and heavily uses analytic tools developed by 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 behaviour.