[Paper Review] Constrained deformations of positive scalar curvature metrics, II

Alessandro Carlotto, Chao Li|arXiv (Cornell University)|Jul 23, 2021

Geometric Analysis and Curvature Flows63 references15 citations

TL;DR

This paper establishes that spaces of positive scalar curvature (PSC) metrics on compact 3-manifolds with boundary—subject to mean-convex, weakly mean-convex, or minimal boundary conditions—are either empty or contractible. Using constrained deformations via singular Ricci flow and equivariant techniques, the authors prove weak homotopy equivalence to doubling metrics, then show these doubling spaces are weakly contractible, extending recent results on PSC metrics in 3D. The key contribution is the homotopy triviality of these constrained PSC metric spaces, with implications for initial data sets in general relativity.

ABSTRACT

We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean-convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.

Motivation & Objective

• To determine the homotopy type of spaces of positive scalar curvature (PSC) metrics on compact 3-manifolds with boundary under boundary mean curvature constraints.
• To extend the contractibility result for PSC metrics from closed 3-manifolds to manifolds with boundary under geometric constraints.
• To establish that spaces of PSC metrics with mean-convex, weakly mean-convex, or minimal boundary are either empty or weakly contractible.
• To apply these results to the topology of asymptotically flat initial data sets in general relativity, particularly under dominant energy and outer trapping conditions.

Proposed method

• Utilizes weak homotopy equivalence to relate constrained PSC metric spaces to the space of doubling metrics (R>0,D), which are PSC metrics extendable via reflection across the boundary.
• Applies singular Ricci flow with surgery in the equivariant setting, adapting Bamler-Kleiner’s framework to preserve symmetry and boundary conditions.
• Employs backward-in-time induction on the singular Ricci flow to construct partial homotopies that preserve PSC and doubling structure.
• Introduces reflexive partial homotopies and constructs equivariant deformations using exponential maps and symmetric embeddings near the fixed point set of the reflection.
• Uses equivariant extension and removal techniques for 3-disks in the flow’s singular structure, ensuring invariance under the involution and compatibility with PSC-conformal metrics.
• Relies on refined topological tools such as normal injectivity radius estimates and reflexive conformal metric constructions to maintain geometric control during deformation.

Experimental results

Research questions

• RQ1 Are spaces of PSC metrics on compact 3-manifolds with boundary, constrained by mean-convex or minimal boundary conditions, contractible when non-empty?
• RQ2 Can the homotopy type of such constrained PSC metric spaces be reduced to that of doubling metrics via controlled deformation?
• RQ3 How does the singular Ricci flow framework adapt to equivariant settings with boundary symmetry to prove contractibility?
• RQ4 What topological implications arise for initial data sets in general relativity when PSC and boundary mean curvature constraints are imposed?
• RQ5 To what extent can the techniques of partial homotopy and backward-in-time induction be extended to manifolds with boundary and involutive symmetries?

Key findings

• The space of PSC metrics with mean-convex boundary (R>0,H>0) is either empty or weakly contractible.
• The space of PSC metrics with weakly mean-convex boundary (R>0,H≥0) is either empty or weakly contractible.
• The space of PSC metrics with minimal boundary (R>0,H=0) is either empty or weakly contractible.
• The space of doubling metrics (R>0,D) is either empty or weakly contractible, serving as a homotopy model for the constrained PSC spaces.
• The constrained PSC metric spaces are weakly homotopy equivalent to the doubling metric space, establishing a topological reduction via controlled boundary straightening.
• The results imply that the moduli space of asymptotically flat initial data sets satisfying dominant energy and outer trapping conditions has trivial higher homotopy groups when constrained by PSC and boundary curvature conditions.