# School of Mathematics

## Progress in the theory of CMC surfaces in locally homgeneous 3-manifolds X

## The embedded Calabi-Yau problem for minimal surfaces of finite genus

Abstract: We will explain how to prove properness of a complete embedded minimal surface in Euclidean three-space, provided that the surface has finite genus and countably many limit ends (and possibly compact boundary).

This is joint work with William H. Meeks and Antonio Ros.

## Generic uniqueness of expanders with vanishing relative entropy

Abstract: We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same smooth cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that generically expanders with vanishing relative entropy are unique. This also implies that generically locally entropy minimizing expanders are unique. This is joint work with A. Deruelle.

## Locating Minimal Surfaces in Geometrostatic Manifolds

## Recent gluing constructions in Geometry and the gluing methodology

Abstract: I will first concentrate on doubling gluing constructions for minimal surfaces, including a recent construction for free boundary minimal surfaces in the unit ball (with D. Wiygul: arXiv:1711.00818).

I will then discuss the Linearized Doubling methodology and its applications so far (J. Differential Geom. 106:393-449, 2017; and with P. McGrath: arXiv:1707.08526),

and some further ongoing work expanding the scope of these methods to new cases.

## Sinh-Gordon equation and application to the geometry of CMC surfaces.

## A Plateau problem at infinity in H2xR

Abstract: We give some existence and non-existence results for minimal annuli in H2xR whose data at infinity are given.

## Progress on existence of minimal surfaces

Abstract: I will survey the recent progress on the existence problem for minimal hypersurfaces and then point for some new directions. This is joint work with Fernando Marques.