Compact embedded minimal surfaces in $\mathbb{S}^2 \times \mathbb{S}^1$

Abstract

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in S 2 × S 1 (r), for arbitrary radius r. We illustrate it by obtaining some periodic minimal surfaces in S 2 × R via conjugate constructions. The resulting surfaces can be seen as the analogy to the Schwarz P-surface in these homogeneous 3-manifolds.

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

“…We will focus on the simply connected three-manifold S 2 × R with non-negative Ricci curvature, where we will overtake the problem of determining the possible topological types of compact embedded H-surfaces. Similar problems have been considered in the literature, e.g., Lawson [5] showed that a compact surface can be minimally embedded in the three-sphere S 3 if and only if it is orientable, and the second author extended this result to the larger family of Berger spheres [16]; moreover, the authors and Plehnert [9] showed that a compact surface can be minimally embedded in a Riemannian product S 2 × S 1 (r) if and only if it has even Euler characteristic. Although the only compact minimal surfaces immersed in S 2 × R are the horizontal slices S 2 × {t 0 }, t 0 ∈ R, in the non-compact case Hoffman, Traizet and White [2] proved the existence of embedded minimal surfaces in S 2 × R with two helicoidal ends and arbitrary genus.…”

Minimal hypersurfaces in ℝn×Sm

“…In this section we will show existence of some invariant constant mean curvature hypersurfaces by studying the solutions of equation (2), with h a positive constant. Of course the equation is considerably more complicated than equation (8). Moreover the clear description obtained in Theorem 1.1 will not be possible in this case.…”

“…But moreover it follows from the Lemma 2.2 and Lemma 2.3 that the local maxima and minima stay bounded away from π and 0 (respectively). If the values of p at the local extrema also stay bounded away π/2 it is elementary and easy to see from equation (8) that the distance between consecutive extrema of p will have a positive lower bound and therefore p would be defined for all x > z. The only possibility left would be that there exists x 0 > 0 such that lim x→x 0 p(x) = π/2.…”

“…And in the last few years there has been great interest in the study of minimal surfaces in 3-dimensional Riemannian products: see for instance the articles by H. Rosenberg [14], W.H. Meeks and H. Rosenberg [9] and J. M. Manzano, J. Plehnert and F. Torralbo [8]. In higher dimensional manifolds constructions are also abundant.…”

A geometrical correspondence between maximal surfaces in anti-De Sitter space–time and minimal surfaces inH2×R

“…Among those spaces, there are three homogeneous Riemannian manifolds: R 3 , where the classical theory of minimal surfaces has been developed, and S 2 × R and H 2 × R, where many authors have been actively working. Giving a complete list of references on the subject is far from being possible so we will only mention a few of them: Nelli and Rosenberg [12] proved a Jenkins-Serrin-type theorem in H 2 ×R, Hauswirth [6] constructed minimal examples of Riemann type, Sá Earp and Tobiana [18] investigated the screw motion invariant surfaces in H 2 ×R, Daniel [4] and Hauwirth, Sá Earp and Tobiana [7] showed, independently, the existence of an associated family of minimal immersions for simply connected minimal surfaces in S 2 ×R and H 2 ×R, Urbano and the author [20] tackled a general study of minimal surfaces in S 2 × S 2 with applications to S 2 × R, very recently Manzano, Plehnert and the author [9] constructed orientable and non-orientable even Euler characteristic embedded minimal surfaces in the quotient S 2 × S 1 and Martín, Mazzeo and Rogríguez [10] constructed the first examples of complete, properly embedded minimal surfaces in H 2 × R with finite total curvature and positive genus.…”