Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres

Abstract

We use equivariant bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

Equivariant Bifurcation in Geometric Variational Problems

“…In this section, we apply our abstract equivariant bifurcation results (Theorems 4.3 and 4.5) to the geometric variational problem of constant mean curvature (CMC) embeddings. Bifurcation phenomena for 1-parameter families of CMC embeddings have been studied in the last years by several authors, see, e.g., [3,6,16,18]. We will state and prove general bifurcation results for CMC embeddings (Theorems 5.4 and 5.8) and discuss how some explicit bifurcation examples can be reobtained from these general results.…”

Rigidity and Bifurcation Results for CMC Hypersurfaces in Warped Product Spaces

“…This section is devoted to introduce roughly the basic framework about bifurcation theory needed for the rest of this paper. For details, we refer the readers to [10,2] and the references therein.…”

“…Now, we are ready to determine the first variation of our functional F λ . For this, consider X a normal variation and f its associated function satisfying (2). Denote by…”

“…Recently, Ramírez-Ospina [8] proved the existence of uncountably many unit volume constant scalar curvature metrics in product manifolds T k ×M where T k is a flat k-torus and M is a compact manifold. Among all these works, we point out one authored by Alías and Piccione [2], where it is shown the existence of an infinite sequence of isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford torus.…”

Delaunay-Type Hypersurfaces in Cohomogeneity One Manifolds

“…Similar constructions of rotationally symmetric CMC hypersurfaces exist in S n , R n and H n . In particular, those in S n are compact and can be understood as bifurcating branches from the family of CMC Clifford tori, see [4,33]. As suggested by Pacard [35], "it is then a natural question to investigate the existence of Date: July 13, 2015.…”