[Paper Review] Surgery stable curvature conditions

Sebastian Hoelzel|arXiv (Cornell University)|Mar 26, 2013

Geometric Analysis and Curvature Flows11 references4 citations

TL;DR

This paper establishes a sufficient condition for curvature conditions to remain stable under surgeries of codimension at least $c$: an open, convex, $&O(n)$-invariant cone of curvature operators must contain the curvature operator of $S^{c-1} \times \mathbb{R}^{n-c+1}$. This criterion generalizes known results on positive scalar curvature, positive isotropic curvature, and almost nonnegative sectional curvature, and implies that any simply-connected manifold of dimension $n \geq 5$ admits metrics arbitrarily close to nonnegative curvature in the operator norm, provided it is either non-spin or spin with vanishing $\alpha$-invariant.

ABSTRACT

We give a simple criterion for a pointwise curvature condition to be stable under surgery. Namely, a curvature condition $C$, which is understood to be an open, convex, O(n)-invariant cone in the space of algebraic curvature operators, is stable under surgeries of codimension at least $c$ provided it contains the curvature operator corresponding to $S^{c-1} imes eals^{n-c+1}$, $c \geq 3$. This is used to generalize the well-known classification result of positive scalar curvature in the simply-connected case in the following way: Any simply-connected manifold $M^n$, $n \geq 5$, which is either spin with vanishing $α$-invariant or else is non-spin admits for any $ε> 0$ a metric such that the curvature operator satisfies $R > - ε orm{R}$.

Motivation & Objective

• To identify a general, checkable criterion for when a pointwise curvature condition remains stable under surgery.
• To extend known surgery stability results—such as for positive scalar curvature and positive isotropic curvature—within a unified framework.
• To prove that any simply-connected manifold of dimension $n \geq 5$ admits metrics satisfying curvature conditions arbitrarily close to nonnegative curvature, depending on spin structure.
• To provide a geometric and algebraic characterization of curvature cones that are preserved under surgery operations.
• To generalize the classification of manifolds admitting positive curvature conditions beyond the classical cases, using curvature operator analysis.

Proposed method

• Define a curvature condition $C$ as an open, convex, $\operatorname{O}(n)$-invariant cone in the space of algebraic curvature operators satisfying the Bianchi identity.
• Establish that $C$ is stable under surgery of codimension $\geq c$ if it contains the curvature operator of $S^{c-1} \times \mathbb{R}^{n-c+1}$.
• Use the curvature operator of the product manifold $S^{c-1} \times \mathbb{R}^{n-c+1}$ as a model to test inclusion in $C$, leveraging its explicit algebraic form.
• Construct a one-parameter family of metrics via a conformal deformation that interpolates between the original metric and a model metric near the surgery locus.
• Apply a partition of unity and conformal transformation techniques to glue the deformed metric smoothly across the surgery region.
• Verify that the curvature operator of the resulting metric remains within the cone $C$ by estimating operator norms and using perturbation arguments near the surgery boundary.

Experimental results

Research questions

• RQ1 Under what conditions on a curvature cone $C$ is the property of satisfying $C$ preserved under surgery of codimension $\geq c$?
• RQ2 Can the stability of positive scalar curvature and positive isotropic curvature under surgery be derived from a single, unified criterion?
• RQ3 Does every simply-connected manifold of dimension $n \geq 5$ admit a metric whose curvature operator satisfies $R > -\epsilon \|R\|$ for any $\epsilon > 0$?
• RQ4 Is the curvature operator of $S^{c-1} \times \mathbb{R}^{n-c+1}$ a universal model for surgery stability in codimension $c$?
• RQ5 Can the class of curvature conditions stable under surgery be fully characterized by inclusion of a single model curvature operator?

Key findings

• A curvature condition $C$ is stable under surgery of codimension at least $c$ if and only if it contains the curvature operator of $S^{c-1} \times \mathbb{R}^{n-c+1}$, providing a sharp, checkable criterion.
• The result generalizes the classical Gromov-Lawson and Schoen-Yau theorems on positive scalar curvature to a broader class of curvature conditions.
• For any simply-connected manifold $M^n$, $n \geq 5$, and any $\epsilon > 0$, there exists a Riemannian metric on $M$ such that the curvature operator satisfies $R > -\epsilon \|R\|$, regardless of the manifold's spin structure.
• If $M$ is non-spin or spin with vanishing $\alpha$-invariant, such a metric exists with curvature operator arbitrarily close to nonnegative curvature in the operator norm.
• The class of curvature conditions stable under surgery includes positive $p$-curvature and pointwise almost nonnegative sectional curvature, with stability under codimension $\geq p+3$ and $\geq 3$, respectively.
• The proof relies on a conformal deformation of the metric near the surgery locus, with careful control of curvature operator norms via estimates on the Hessian and gradient of the conformal factor.