Constructions of periodic minimal surfaces and minimal annuli in Sol$_3$

Christophe Desmonts

Vol. 276 (2015), No. 1, 143–166

DOI: 10.2140/pjm.2015.276.143

Abstract

We construct two one-parameter families of minimal properly embedded surfaces in the Lie group Sol${}_3$ using a Weierstrass-type representation. These surfaces are not invariant by a one-parameter group of ambient isometries. The first one can be viewed as a family of helicoids, and the second one as a family of minimal annuli called catenoids. Finally we study limits of these catenoids, and in particular we show that one of these limits is a new minimal entire graph.

Keywords

periodic minimal surfaces, minimal annuli, Sol$_3$

Mathematical Subject Classification 2010

Primary: 53A10

Milestones

Received: 22 January 2014

Revised: 1 August 2014

Accepted: 12 August 2014

Published: 1 July 2015