Surfaces in $\mathbb{R}^3_{+}$ with the same Gaussian curvature induced by the Euclidean and hyperbolic metrics

Nilton Barroso and Pedro Roitman

Vol. 275 (2015), No. 1, 19–37

DOI: 10.2140/pjm.2015.275.19

Abstract

We show how to construct infinitely many immersions into the upper half-space such that the Gaussian curvatures induced from the ambient Euclidean and hyperbolic metrics coincide. We show how these immersions are related geometrically to classical minimal surfaces in Euclidean space and timelike minimal surfaces in Minkowski space.

Keywords

minimal surfaces, Euclidean geometry, hyperbolic geometry, Gaussian curvature, Monge–Ampère equations

Mathematical Subject Classification 2010

Primary: 53A10, 53A35

Secondary: 35L70, 35J96

Milestones

Received: 16 May 2013

Revised: 28 July 2014

Accepted: 13 November 2014

Published: 12 April 2015