Degree theory for 4-dimensional asymptotically conical gradient expanding solitons

TL;DR

This paper introduces a localized, integer-valued expander degree $\deg_{\exp}(X)$ for 4-dimensional orbifolds with boundary to study asymptotically conical gradient expanding solitons. By embedding AC solitons into a larger, non-gradient soliton space and employing DeTurck-type gauges, the authors prove existence results for solitons asymptotic to cones with non-negative scalar curvature, including cones over $S^3$ and $S^3/\Gamma$, with the expander degree acting as a topological obstruction/guide. They compute $\deg_{\exp}(D^4)=\deg_{\exp}(D^4/\Gamma)=1$ and show that the expander degree vanishes for certain exotic 4-disks, linking smooth structure to soliton existence. A key outcome is that the gradient subset of AC solitons forms unions of connected components within the AC soliton space, and under suitable deformations, the gradient property is preserved; the framework relies on refined elliptic estimates, weighted function spaces, and a robust compactness theory for solitons with generalized cone ends. Overall, the work provides a foundational tool for constructing expanding solitons with prescribed conical asymptotics and offers new insights into soliton-geometry/topology interactions relevant to 4D Ricci flow with surgery.

Abstract

We develop a new degree theory for 4-dimensional, asymptotically conical gradient expanding solitons. Our theory implies the existence of gradient expanding solitons that are asymptotic to any given cone over $S^3$ with non-negative scalar curvature. We also obtain a similar existence result for cones whose link is diffeomorphic to $S^3/Γ$ if we allow the expanding soliton to have orbifold singularities. Our theory reveals the existence of a new topological invariant, called the expander degree, applicable to a particular class of compact, smooth 4-orbifolds with boundary. This invariant is roughly equal to a signed count of all possible gradient expanding solitons that can be defined on the interior of the orbifold and are asymptotic to any fixed cone metric with non-negative scalar curvature. If the expander degree of an orbifold is non-zero, then gradient expanding solitons exist for any such cone metric. We show that the expander degree of the 4-disk $D^4$ and any orbifold of the form $D^4/Γ$ equals 1. Additionally, we demonstrate that the expander degree of certain orbifolds, including exotic 4-disks, vanishes. Our theory also sheds light on the relation between gradient and non-gradient expanding solitons with respect to their asymptotic model. More specifically, we show that among the set of asymptotically conical expanding solitons, the subset of those solitons that are gradient forms a union of connected components.

Theorems & Definitions (226)

• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.8
• Theorem 1.9
• Theorem 1.11
• Theorem 1.13
• Theorem 1.14
• Conjecture 1.17
• Theorem 2.3