One-sided complete stable minimal surfaces

Abstract

We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of R 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact twosided index one minimal surfaces in non-negatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface.

Stable periodic constant mean curvature surfaces and mesoscopic phase separation

“…Thus, it follows from (1) that Σ is totally geodesic. If Σ is connected, then Ros [25] proves that the genus of Σ is 3 (for earlier partial results see [6,20,19,30]). …”

“…It was proved in [25] that the genus of a closed stable surface in a 3-manifold with nonnegative Ricci curvature is at most 3. Pedrosa [17] solved the isoperimetric problem in the product S 2 × R of a round sphere and a line.…”

“…If k = 0, then Barbosa and do Carmo [4] proved that the only stable surfaces are round spheres. For 1 k 3 it is known that, if Σ is stable, then it must be either a union of planar surfaces or a connected surface of genus smaller than or equal to 3 (Ros [25]). This result is sharp for k = 3 as Ross [27] has proved that the classical Schwarz P minimal surface (of genus 3) in the simple cubic torus R 3 /Z 3 is stable.…”

“…In this paper we will first use test functions obtained from harmonic 1-forms on the surface (see Palmer [16] and Ros [25] for other applications of harmonic forms to stability problems). We will also construct test functions by using conformal maps between suitable pieces of Σ and the 2-sphere.…”

“…k = 3) the above gives a new proof of the fact that closed stable constant mean curvature surfaces in a flat 3-torus have genus 3 (Ros [25]). However, this result can be qualitatively improved in the embedded case, by combining all the power of Theorem 2 with topological properties of constant mean curvature embeddings.…”