Regularity of the analytic torsion form on families of normal coverings

Bing Kwan So and GuangXiang Su

Vol. 291 (2017), No. 1, 149–181

DOI: 10.2140/pjm.2017.291.149

Abstract

We prove the smoothness of the $L^2$-analytic torsion form for fiber bundles with positive Novikov–Shubin invariant. We do so by generalizing the arguments of Azzali, Goette and Schick to an appropriate Sobolev space, and proving that the Novikov–Shubin invariant is also positive in the Sobolev setting, using an argument of Alvarez Lopez and Kordyukov.

Keywords

$L^2$-analytic torsion form, Novikov-Shubin invariant, Sobolev space, superconnection

Mathematical Subject Classification 2010

Primary: 58J52

Secondary: 58J35

Milestones

Received: 9 June 2014

Revised: 16 July 2016

Accepted: 7 October 2016

Published: 20 August 2017