Isoparametric surfaces in $E(\kappa,\tau)$-spaces

Abstract

We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and surfaces with constant mean curvature and vanishing Abresch-Rosenberg differential.2010 Mathematics Subject Classification. Primary 53A10; Secondary 53C30, 53B25.

Hypersurfaces of Constant Higher Order Mean Curvature in $M\times\mathbb{R}$

“…Thus, Σ is strictly convex at o. In addition, by the identities (12), at any point of Σ −{o}, all the principal curvatures are positive. Therefore, Σ is strictly convex.…”

“…Example 3. In [12], the authors constructed isoparametric families of parallel surfaces with nonzero constant principal curvatures in E(k, τ) spaces satisfying k − 4τ 2 = 0. (Those include the products H 2 × R and S 2 × R, the Heisenberg space Nil 3 , the Berger spheres, and the universal cover of the special linear group SL 2 (R)).…”

Submanifold geometry in symmetric spaces of noncompact type

“…This motivates the problem of classifying homogeneous hypersurfaces or, equivalently, cohomogeneity one actions up to orbit equivalence, in specific Riemannian manifolds, mainly in those with large isometry group. Such classification is known, for example, for Euclidean and real hyperbolic spaces (as a consequence of Segre's [82] and Cartan's [24] works on isoparametric hypersurfaces, see §4.3), irreducible symmetric spaces of compact type [66], and simply connected homogeneous 3-manifolds with 4-dimensional isometry group [48]. Below we focus on the classification problem in symmetric spaces of noncompact type, and refer the reader to [5, §6] and [8, §2.9.3 and Chapters 12-13] for more information on cohomogeneity one actions.…”

“…In spaces of nonconstant curvature, the problem becomes very complicated. Apart from the results we will review below, there is a classification on complex projective spaces CP n , n = 15 [45], quaternionic projective spaces HP n , n = 7 [47], the product S 2 × S 2 [87], and simply connected homogeneous 3-manifolds with 4-dimensional isometry group [48], such as the products S 2 × R, RH 2 × R, the Heisenberg group Nil 3 or the Berger spheres. Interestingly, in all the cases mentioned so far (as well as in the rest of examples presented in this paper) an isoparametric hypersurface is always an open subset of a leaf of an isoparametric foliation of codimension one that fills the whole ambient space.…”

Helicoids and Catenoids in $M\times\mathbb{R}$

“…It is well known that, when M is one of the constant sectional curvature space forms, such a family g s In this context, Damek-Ricci spaces contrast with the E(k, τ )-spaces with k − 4τ 2 = 0, i.e., the simply connected 3-homogeneous manifolds with isometry group of dimension 4: The products H 2 × R and S 2 × R (τ = 0), the Heisenberg space Nil 3 (k = 0, τ = 0), the Berger spheres (k > 0, τ = 0), and the universal cover of the special linear group SL 2 (R) with some special left-invariant metrics (k < 0, τ = 0). Indeed, in [9], the authors classified all isoparametric hypersurfaces of these spaces, and none of them is spherical. Therefore, for any E(k, τ )-space, k − 4τ 2 = 0, there exist vertical catenoids in E(k, τ ) × R, and none of them is rotational.…”