Regularity of analytic torsion form on families of normal coverings

TL;DR

This paper establishes the smoothness of the L^2-analytic torsion form on certain fiber bundles with non-compact fibers, extending previous methods to Sobolev spaces and ensuring the Novikov-Shubin invariant remains positive.

Contribution

It generalizes existing arguments to Sobolev spaces and proves the positivity of the Novikov-Shubin invariant in this setting.

Findings

Proves smoothness of L^2-analytic torsion form on specific fiber bundles.

Extends arguments to Sobolev spaces for non-compact fibers.

Shows Novikov-Shubin invariant remains positive in Sobolev settings.

Abstract

We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev settings, using an argument of Alvarez Lopez-Kordyukov.