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.
Abstract: We give some existence and non-existence results for minimal annuli in H2xR whose data at infinity are given.
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.
Abstract: In this talk we show that given any regular cone with entropy less than that of round cylinder, all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class. This is joint work with J. Bernstein.
Based on joint works with Xin Li, Noam Solomon and Jiapeng Zhang.