Triply periodic constant mean curvature surfaces

Abstract

Given a closed flat 3-torus N , for each H > 0 and each non-negative integer g, we obtain area estimates for closed surfaces with genus g and constant mean curvature H embedded in N . This result contrasts with the theorem of Traizet [33], who proved that every flat 3-torus admits for every positive integer g with g = 2, connected closed embedded minimal surfaces of genus g with arbitrarily large area. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42.We now recall several key notions and theorems from [20] that we will apply in the next section. Definition 2.1 Let M be an H-surface, possibly with boundary, in a complete oriented Riemannian 3-manifold N . 1. M is an H-disk if M is diffeomorphic to a closed disk in the complex plane. 2. |A M | denotes the norm of the second fundamental form of M .3. The radius of a Riemannian n-manifold with boundary is the supremum of the intrinsic distances of points in the manifold to its boundary.The next two results are contained in [20], see also [18,19,21,22,23].Theorem 2.2 (Radius Estimates) There exists an R ≥ π such that any H-disk in R 3 has radius less than R/H.

Gap phenomena for constant mean curvature surfaces

“…In the specific case when the ambient space is $$\mathbb {R}^3$$, we can use the recent work of Meeks and Tinaglia [17, 18] about embedded CMC surfaces in $$\mathbb {R}^3$$. As consequence of their work, we know that round spheres are the only complete simply connected surfaces embedded in $$\mathbb {R}^3$$ with nonzero CMC.…”