Slice-Quaternionic Hopf Surfaces

Abstract

We investigate slice-quaternionic Hopf surfaces. In particular, we construct new structures of slice-quaternionic manifold on S 1 × S 7 , we study their group of automorphisms and their deformations.In the following cases, the quotient of H 2 \ {(0, 0)} by the subgroup generated by f yields a structure of slice-quaternionic Hopf surface: Case A.: Case A.1.: when λ = 0, α = β ∈ H with 0 < |α| < 1; Case A.2.: when λ = 0, α, β ∈ H with 0 < |α| ≤ |β| < 1 and α = β; Case A.3.: when λ ∈ H with λ = 0, p = 1, α = β ∈ H with 0 < |α| < 1; Case B.: when λ ∈ H with λ = 0, p ∈ N with p > 1, β ∈ R with 0 < |β| < 1, and α = β p .Remark. We wonder whether other slice-quaternionic structures on S 1 × S 7 may be constructed; see Remark 2.3.The automorphism groups of the above slice-quaternionic structures are investigated in Theorem 3.1. Notice that, in general, a slice-quaternionic structure does not induce a holomorphic structure; compare also Remark 2.2. (Note indeed that (where I, J are orthogonal complex structures on R 4 .) Therefore the slice-quaternionic Hopf surfaces in case (B) do not underlie a complex Calabi-Eckmann structure.We prove the following result. Theorem 3.1. Let X = H 2 \ {(0, 0)} f be a slice-quaternionic Hopf surface, where f is as in equation (0.1). The dimension of the group of automorphisms of X is as follows: Case A.1.: dim R Aut(X) ∈ {8, 16}; Case A.2.: dim R Aut(X) ∈ {4, 6, 8}; Case A.3.: dim R Aut(X) ∈ {4, 8}; Case B.: dim R Aut(X) = 5. Finally, we provide families of slice-quaternionic Hopf surfaces, connecting cases (A.1) and (A.3), respectively cases (A.2.1) and (B), see Section §4.Remark 0.1. As suggested by the anonymous Referee, we observe that analogous constructions can be performed to obtain slice-quaternionic Hopf manifolds of higher dimension, and that similar techniques might be possibly developed for the study of other slice-quaternionic manifolds, for Suppose now α ∈ R, and that λ ∈ R. By imposing the slice regularity of the left-hand side, we get that: a h,k = 0 and b h,k = 0 for (h, k) ∈ {(1, 0), (0, k) : k ≥ 1}. Therefore we are reduced to:We equal: firstly, the coefficients in z; secondly, the coefficients in µ. We are reduced to: α · a 1,0 = a 1,0 · α + b 1,0 · λ , λ · a 1,0 + α · a 0,1 = a 0,1 · α + b 0,1 · λ , α k · a 0,k = a 0,k · α + b 0,k · λ , for k > 1 , α · b 1,0 = b 1,0 · α , λ · b 1,0 + α · b 0,1 = b 0,1 · α ,

The Harmonicity of Slice Regular Functions

“…We hope that this paper can provide new ideas for studying slice regular functions and their "harmonic properties" on slice regular quaternionic manifolds recently introduced by Bisi-Gentili in [12] and Angella-Bisi in [5].…”

On a quaternionic Picard theorem

“…Maybe this work can be of some inspiration in studying hyperbolic quaternionic slice regular manifolds. Indeed recently many examples of quaternionic slice regular manifolds have been introduced; see for example [2], [1].…”

Log-biharmonicity and a Jensen formula in the space of quaternions

“…After that, in Corollaries 3.10 and 3.12, we give upper bounds on the number of zeros of a slice regular function under some additional hypotheses. The following Corollaries 3.13 and 3.14 give formulas for the computation of some integrals over H. The results contained in this paper will be further developed in the understanding of harmonic analysis on quaternionic manifolds (see [9], [13] and [12]).…”