Generalized $S^1$-stability theorem

TL;DR

This paper proves a new stability theorem linking the Yamabe invariants of certain compact manifolds with boundary and their products with a circle, extending previous conjectures using equivariant $$-bubbles.

Contribution

It generalizes the $S^1$-stability conjecture to manifolds with boundary for dimensions 3, 5, and 6 using a novel technique.

Findings

Yamabe invariant positivity is equivalent for $M^n$ and $M^n \times S^1$ in specified dimensions.

The equivariant $$-bubbles technique is effective for manifolds with boundary.

Extension of Rosenberg's $S^1$-stability conjecture to new classes of manifolds.

Abstract

We use the equivariant $\mu$-bubbles technique to prove that for any compact manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is positive. This generalized the $S^1$-stability conjecture of Rosenberg to compact manifolds with boundary.