Spruck y Xiao demostraron en 2018 que todo solitón por traslación del FCM en \( \mathbb{R}^3\) que sea un grafo completo tiene que ser convexo \(K \geq 0\). Nosotros daremos una demostración alternativa de ese teorema, basada en un trabajo conjunto con D. Hoffman y B. White.

We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. This is a joint work with D. Hoffman and B. White.

In this article we prove that a connected, properly embedded translating soliton with uniformly bounded genus on compact sets which is \(C^1\)-asymptotic to two parallel planes outside a cylinder, must coincide with the grim reaper cylinder.

We use Morse theory and the Alexadrov reflection principle to obtain topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space. This is a joint work with K. Smoczyk and A. Savas-Halilaj.

We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of the horizontal catenoid.
This is a joint work with Rafe Mazzeo and Magdalena Rodríguez.

We prove that, given S an open oriented surface, then there exists a complete, proper, area minimizing embedding $f:S→\mathbb{H}^3$. The main tool in the proof of the above result is a sort of bridge principle at infinity for properly embedded area minimizing surfaces in hyperbolic three space.
This is a joint work with Brian White.

Dado D un dominio en una 3-variedad y M una superficie abierta, se dice que D es un dominio de Calabi-Yau para M si no se puede construir una inmersión completa y propia de M en D que tenga curvatura media acotada. Nosotros veremos que si M tiene un final anular, entonces toda tres variedad admite un dominio de Calabi-Yau para M.

Discutiremos algunos resultados recientes relacionados con la conjetura de Calabi-Yau para superficies minimales embebidas de R^3