## Translating solitions of the mean curvature flow

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Pérez García, Jesús##### Editorial

Universidad de Granada

##### Departamento

Universidad de Granada. Departamento de Geometría y Topología##### Materia

Curvatura Superficies (Matemáticas) Solitones Singularidades (Matemáticas)

##### Materia UDC

(043.2) 51 517 6305

##### Date

2016##### Fecha lectura

2016-12-16##### Referencia bibliográfica

Pérez García, J. Translating solitions of the mean curvature flow. Granada: Universidad de Granada, 2016. [http://hdl.handle.net/10481/44530]

##### Sponsorship

Tesis Univ. Granada. Programa Oficial de Doctorado en: Matemáticas; This research was supported by Ministerio de Economía y Competitividad (FPI grant, BES-2012-055302), by MICINN-FEDER grant number MTM2011-22547 and by MINECO-FEDER grant number MTM2014-52368-P.##### 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.