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.
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.
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.