Compact embedded surfaces with constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$

Abstract

We obtain compact orientable embedded surfaces with constant mean curvature 0 < H < 1 2 and arbitrary genus in S 2 × R. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature 1 2 tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in S 2 × R, H 2 × R, and R 3 with bounded height and enjoying the symmetries of certain tessellations of S 2 , H 2 , and R 2 by regular polygons.

Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times\mathbb{R}$

“…The construction of these genus 1 minimal k-noids is based on a conjugate technique, in the sense of Daniel [2] and Hauswirth, Sa Earp and Toubiana [6]. Conjugation has been a fruitful technique to obtain constant mean curvature surfaces in H 2 × R and S 2 × R, see [7,8,9,11,13,14,16,17,18] and the references therein. We begin by considering a solution to an improper Dirichlet problem [12,15] in H 2 × R over an unbounded geodesic triangle ∆ ⊂ H 2 , a so-called Jenkins-Serrin problem.…”