2024/03/01 by Michael Heins, Annika Moucha, Oliver Roth
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.4153/s0008414x2400021x
Abstract The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement of the “complexified unit circle” \(z,w) ∈ \widehat \mathbb C2 \colon z ⋅ w = 1\ . We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric 2F1 functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed Möbius invariant subspaces of eigenspaces of the canonical Laplacian of Ω . Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real-analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.