The Mean Curvature Flow (MCF) describes the evolution of hypersurfaces in Euclidean space, driven by their mean curvature, which tends to smooth out geometric irregularities over time. However, singularities inevitably develop during the flow, marking critical points where the smooth evolution ceases. In this talk, we will examine the formation of singularities in MCF, focusing on the crucial role of tangent flows in their analysis. Tangent flows, which emerge as blow-up limits near singularities, often exhibit self-similar structures. We will highlight how the mean curvature flow produces a specific type of tangent flow at the first singularity, preserving notable geometric and topological properties of the compact initial data. This presentation is based on an ongoing work with David Hoffman and Brian White.

We complete the classification of semigraphical translators for the mean curvature flow in \(\mathbb{R}^3\), a study initiated by Hoffman, Martín, and White. Specifically, we demonstrate that no solutions exist to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values—a configuration previously referred to as the "Yeti". Furthermore, we establish the uniqueness of pitchfork and helicoid translators. Our proofs use Morse-Radó theory for translators and an angular maximum principle. This is joint work with Mariel Sáez, Raphael Tsiamis, and Brian White.

We say that a surface is semigraphical if it is properly embedded, and, after removing a discrete collection of vertical lines, it is a graph. In this talk, we provide a nearly complete classification of semigraphical translators.

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