Conjugate Plateau constructions in product spaces

Abstract

This survey paper investigates, from a purely geometric point of view, Daniel's isometric conjugation between minimal and constant mean curvature surfaces immersed in homogeneous Riemannian three-manifolds with isometry group of dimension four. On the one hand, we collect the results and strategies in the literature that have been developed so far to deal with the analysis of conjugate surfaces and their embeddedness. On the other hand, we revisit some constructions of constant mean curvature surfaces in the homogeneous product spaces S 2 × R, H 2 × R and R 3 having different topologies and geometric properties depending on the value of the mean curvature. Finally, we also provide some numerical pictures using Surface Evolver.

Polyharmonic hypersurfaces into space forms

“…In the case of interest for us we have the following important result: Remark 2.5. We point out that in references [4,6,7] the authors use parameters κ, τ instead of m, ℓ. The relationship between these parameters is κ = 4m, τ = ℓ/2.…”

“…Now, we recall (see [4]) some basic facts about the half-space model. We have an explicit isometry Θ :…”