Sp(2)-invariant expanders and shrinkers in Laplacian flow

TL;DR

This paper classifies and analyzes Sp(2)-invariant expanders and shrinkers in Laplacian flow, revealing their asymptotic behaviors, uniqueness, and proposing conjectures about similar structures in SU(3)-invariant cases, suggesting a potential flow surgery mechanism.

Contribution

It provides a complete classification of Sp(2)-invariant expanders and shrinkers, determines their asymptotic cones, and introduces new conjectures about SU(3)-invariant solutions and flow surgery.

Findings

Sp(2)-invariant expanders form a 1-parameter family and are asymptotically conical.

The asymptotic cone uniquely determines the complete expander up to scale.

New forward-complete end solutions with faster-than-Euclidean volume growth are identified.

Abstract

We show that the complete Sp(2)-invariant expanding solitons for Bryant's Laplacian flow on the anti-self-dual bundle of the 4-sphere form a 1-parameter family, and that they are all asymptotically conical (AC). We determine their asymptotic cones, and prove that this cone determines the complete expander (up to scale). Neither the unique Sp(2)-invariant torsion-free G_2-cone nor the asymptotic cone of the explicit AC Sp(2)-invariant shrinker from arxiv:2112.09095 occurs as the asymptotic cone of a complete AC Sp(2)-invariant expander. We determine all possible end behaviours of Sp(2)-invariant solitons, identifying novel forward-complete end solutions for both expanders and shrinkers with faster-than-Euclidean volume growth. We conjecture that there exists a 1-parameter family of complete SU(3)-invariant expanders on the anti-self-dual bundle of the complex projective plane CP^2 with such asymptotic behaviour. We also conjecture that, in contrast to the Sp(2)-invariant case, there exist complete SU(3)-invariant AC expanders with asymptotic cone matching that of the explicit AC SU(3)-invariant shrinker from arxiv:2112.09095. The latter conjecture suggests that Laplacian flow may naturally implement a type of surgery in which a CP^2 shrinks to a conically singular point, but after which the flow can be continued smoothly, expanding a topologically different CP^2 from the singularity.