A class of minimal submanifolds in spheres

Abstract

We introduce a class of minimal submanifolds M n , n ≥ 3, in spheres S n+2 that are ruled by totally geodesic spheres of dimension n − 2. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions n = 3 and n = 4. In the first case, we have that M 3 must be a S 1 -bundle over a minimal torus T 2 in S 5 and in the second case M 4 has to be a S 2 -bundle over a minimal sphere S 2 in S 6 . In addition, we provide new examples in relation to the well-known Chern-do Carmo-Kobayashi problem since taking the torus T 2 to be flat yields a minimal submanifolds M 3 in S 5 with constant scalar curvature.In several directions, this paper should be considered as a continuation of our work in [8] where a new class of minimal ruled submanifolds M n of Euclidean space R n+2 , n ≥ 3, were studied. These submanifolds lay in codimension two and may be metrically complete regardless the dimension. The rulings are of codimension two in the manifold whereas the rank, that is, the complement of the index of relative nullity, is ρ = 4 (unless n = 3 = ρ) along an open dense subset. If simply-connected, the submanifolds admit a S 1 -parameter family of genuine isometric deformations. Hence, this class of examples should be seen as a new addition to the possible, local or global, classification of Euclidean submanifolds in codimension two that admit genuine isometric deformations; see [8] for a discussion of that open problem.In this paper, we consider a similar construction but for the round sphere as ambient space. We obtain minimal submanifolds M n in S n+2 , n ≥ 3, with similar properties as the ones in the Euclidean space. Notice that being ruled now means that the submanifold carries a foliation by (open subsets of) totally geodesic spheres in S n+2 of dimension n−2. If the manifold is simply-connected, by taking the cones in R n+3 of the components in the associated family in S n+2 we obtain a new class of genuinely deformable Euclidean submanifolds in codimension two but, of course, these are not complete.

Minimal submanifolds with nullity in space forms

“…If the manifold is simply connected we show that it allows a one-parameter associate family of equally ruled minimal isometric deformations that are genuine. These results may be considered as a continuation of those in [25] and [26].…”

A representation for pseudoholomorphic surfaces in spheres

“…Let g : L 2 → S 2n , n ≥ 3, be a pseudoholomorphic surface. We constructed in [8] an associated ruled minimal submanifold F g : M 2n−2 → S 2n by attaching at each point of g the totally geodesic (2n − 4)-sphere of S 2n whose tangent space at that point is the fiber of the vector bundle Λ g = (N g 1 ) ⊥ , that is, (p, w) ∈ Λ g → F g (p, w) = exp g(p) w (outside singular points) where exp is the exponential map of S 2n . As above, any w ∈ Λ g can be written as w = n−2 j=1 Re(w j F j ) where w j = u j + iv j , 1 ≤ j ≤ n − 2.…”

“…Next, we describe how to parametrize the minimal ruled submanifolds in spheres with codimension two given in [8] but only when associated to a pseudoholomorphic surface.…”

“…Let g : L 2 → S 2n , n ≥ 3, be a pseudoholomorphic surface. We constructed in [8] an associated ruled minimal submanifold F g : M 2n−2 → S 2n by attaching at each point of g the totally geodesic (2n − 4)-sphere of S 2n whose tangent space at that point is the fiber of the vector bundle Λ g = (N g 1 ) ⊥ , that is,…”

“…As above, any w ∈ Λ g can be written as w = n−2 j=1 Re(w j F j ) where w j = u j + iv j , 1 ≤ j ≤ n − 2. Then, the submanifold F g : M 2n−2 → S 2n in [8] can be parametrized as (u j Re(F j ) − v j Im(F j ))…”