On the spectral sets of Inoue surfaces

Abstract

We study the Inoue surfaces SM with the Tricerri metric and the canonical spin c structure, and the corresponding chiral Dirac operators twisted by a flat C * -connection. The twisting connection is determined by z ∈ C * , and the points for which the twisted Dirac operators D ±z are not invertible are called spectral points. We show that there are no spectral points inside the annulus α −1/4 < |z| < α 1/4 , where α > 1 is the only real eigenvalue of the matrix M that determines SM , and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on SM .

Leafwise flat forms on Inoue-Bombieri surfaces

“…For Fourier series on Heisenberg-type nilmanifolds, see e.g. [AT75, DS84,Ric82] as well as the very recent [HZ20a,HZ20b,RS20] which also apply this to study geometric questions.…”