In the first lecture of this series, we show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time $t\to\pm \infty$, any finite entropy, finite genus, embedded, collapsed translating soliton in $\mathbb{R}^3$ converges to a uniquely determined collection of planes. To show the essential necessity of the above line of reasoning and sharpness of the completeness assumption in the main theorems, we will construct counterexamples which are complete with boundary and for which the theorems fail: The nonlinear problem admits solutions, graphical over half-planes, with non-removable singularities at infinity in the sense that a \emph{continuum} of planes all occur as subsequential limits. This is a joint work with F. Martín and N.M Moller.

In this talk we are going to discuss about new examples of translating solitons with entropy five and six.

In this lecture, we are going to talk about a version of the barrier principle for varifolds at infinity. The main aims of this lecture is to prove the validity of the equality

\( \mathrm{dist}(\partial\Omega,\mathrm{spt}\|\Sigma\| )= \mathrm{dist}(\partial\Omega,\mathrm{spt}\|\partial\Sigma\| ), \)

when \(\Omega\) is an open set in a complete Riemannian manifold \(M\) both with a particular structure and \(\Sigma\) is varifolds with bounded mean curvature satisfies a particular condition. This work was done jointly with Jorge H. de Lira (Universidad Federal do Ceará), Luciano Mari (Universitá degli Studi di Torino) and Adriano A. de Medeiros (Universidade Federal da Paraı́ba).

Acceso a la sala

Dada \(M\) una variedad de Riemann de dimensión \(n\) y \(\Omega\) un dominio en \(M\) con frontera regular a trozos, obtenemos condiciones necesarias para la existencia de grafos completos sobre \(\Omega\) en los siguientes casos: - Grafos mínimos - Grafos de CMC - Grafos de traslación para el flujo por la curvatura media.

Obtenemos un teorema de caracterización del cilindro construido sobre una curva grim reaper como el único solitón de traslación del flujo por la curvatura media en el espacio euclídeo \(R^n\) que es asintótico a dos hiperplanos fuera de un cilindro. Esto generaliza a dimensión arbitraria un resultado previo de Martin-Perez-Savas-Smoczyk para solitones de \(R^3\).