Rigidity Conjecture for the Hexablock Automorphism Group

Establish that G(ℍ) equals Aut(ℍ), where G(ℍ) is the subgroup of automorphisms of the hexablock ℍ consisting precisely of the maps T_{ν,χ,ω}(a,x) and T_{ν,χ,F,ω}(a,x) defined by T_{ν,χ,ω}(a,x) = ( ω ξ₂ a √[(1−|z₁|²)(1−|z₂|²)] / (1 − x₁ z̄₁ − x₂ z̄₂ ξ₂ + x₃ z̄₁ z̄₂ ξ₂), τ_{ν,χ}(x) ) and T_{ν,χ,F,ω}(a,x) = ( ω ξ₂ a √[(1−|z₁|²)(1−|z₂|²)] / (1 − x₁ z̄₁ − x₂ z̄₂ ξ₂ + x₃ z̄₁ z̄₂ ξ₂), τ_{ν,χ,F}(x) ), with x=(x₁,x₂,x₃)∈ℰ, ν=−ξ₁ B_{z₁}, χ=−ξ₂ B_{−z̄₂}, B_α(λ)=(λ−α)/(ᾱ λ−1), z₁,z₂∈𝔻, ξ₁,ξ₂∈𝕋, ω∈𝕋, and τ_{ν,χ}, τ_{ν,χ,F}∈Aut(ℰ); here ℍ⊂ℂ⁴ is the hexablock Hartogs domain over the tetrablock ℰ defined by {(a,x)∈ℂ×ℰ : sup_{z₁,z₂∈𝔻} |Ψ_{z₁,z₂}(a,x)| < 1}.

Background

The hexablock ℍ is a Hartogs domain in ℂ⁴ fibered over the tetrablock ℰ⊂ℂ³ and arises from μ-synthesis problems. Biswas–Pal–Tomar constructed a particular subgroup G(ℍ) of automorphisms of ℍ that preserves the tetrablock base via known automorphisms of ℰ and acts on the fiber coordinate a by explicit fractional linear formulas.

They proposed that this subgroup G(ℍ) exhausts the full automorphism group Aut(ℍ). The conjecture asserts a rigidity property: every automorphism of ℍ should be of the explicitly described form T_{ν,χ,ω} or T_{ν,χ,F,ω}, thereby characterizing Aut(ℍ) completely.