Compact minimal surfaces in the Berger spheres

Abstract

In this article, we construct compact, arbitrary Euler characteristic, orientable and non-orientable minimal surfaces in the Berger spheres. Also, we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in S 2 × R, H 2 × R and the Heisenberg group with many symmetries.

Einstein with skew-torsion connections on Berger spheres

“…. This allows us to compute the corresponding Ricci tensor (15) Ric α (X, Y ) = 2(ε(q 2 − 2q + 2) + n + 1)Re(z tw ) + 2nε 2 (q 2 − 2q)ab, which is always symmetric. Thus, the Ricci tensor is a scalar multiple of the metric g ε if and only if ε(q 2 − 2q + 2) + n + 1 = nε(q 2 − 2q), or equivalently if (q − 1) 2 = 2ε + n + 1 ε(n − 1) + 1 = (ε + 1)(n + 1) ε(n − 1) .…”

“…From another point of view, the study of submanifolds immersed in Berger spheres has also been an active research field. Recall, for example, the works on Willmore surfaces [3] and minimal surfaces [15]. Moreover, for suitable scales of the fibers of the Hopf fibration, we can also consider the Berger spheres with Lorentzian signature (see Section 2 for details).…”

Equivariant CR minimal immersions from $$S^3$$S3 into $$\mathbb CP^n$$CPn

“…First we state the following definition of Berger sphere (cf. [9,23]) Definiton 2.1. Let {ω ′ i } be an oriented frame of S 3 with respect to ds 2 0 , and b, c are two positive real numbers.…”

The geometry of stable minimal surfaces in metric Lie groups

“…When trace(A) = 0, we obtain the non-unimodular semidirect products, among which we highlight the hyperbolic space H 3 and the Riemannian product H 2 × R. The geometry of minimal surfaces in homogeneous 3-manifolds of non-constant sectional curvature has been deeply studied in the last decade, specially in the case that the isometry group of the homogeneous manifold has dimension four. To indicate just a few relevant works in this area, we may cite [1,2,3,4,5,6,11,12,17,26,31,33,35]. An outline of the beginning of the theory of constant mean curvature surfaces in homogeneous 3-manifolds with a 4-dimensional isometry group can be consulted in [7,13].…”