In the 1980s Xavier proved that a complete non-planar minimal surface with bounded curvature of $\mathbb{R}^{3}$ can not lie in half-space. In 1990, Hoffman-Meeks proved that this half-space property holds for properly immersed non-planar minimal surfaces of $\mathbb{R}^{3}$ as well. And they went further, proving what is called “the strong half-space theorem” that states that two properly immersed minimal surfaces of $\mathbb{R}^{3}$ intersect unless they are parallel planes. In this talk, I am going to show that a recurrent minimal surface and a complete minimal surface with bounded curvature intersect unless they are parallel planes.  I am also going to present extensions of “half-space theorems” for $H$-surfaces proved by Rosenberg-Schulz-Spruck (2013) and Colombo-Magliaro-Mari-Rigoli (2020).

This is joint work with Luquesio Jorge and Leandro Pessoa.

Zoom invitation is sent to the geometry and topology seminar list. If you are not on the list, please, contact Peter McGrath host to get the link.