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arxiv: 2412.21050 · v2 · submitted 2024-12-30 · 🧮 math.DG · math.AP

Parabolic gap theorems for the Yang-Mills energy

Anuk Dayaprema , Alex Waldron This is my paper

Pith reviewed 2026-05-23 06:41 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords Yang-Mills flowinstantonsgap theoremsdeformation retraction4-spherequaternion-Kähler manifoldsMorrey normYang-Mills energy
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The pith

Connections with Yang-Mills energy below 4π²(|κ| + 2) on an SU(r)-bundle over the 4-sphere deformation-retract onto instantons under the flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the Yang-Mills flow supplies a deformation retraction from the space of sufficiently low-energy connections to the space of instantons. On the 4-sphere this holds for energy less than 4π²(|κ| + 2) and yields a simplified proof of Taubes path-connectedness. On compact quaternion-Kähler manifolds with positive scalar curvature the same retraction works for pseudo-holomorphic connections whose sp(1) curvature component has small Morrey norm. A separate result shows that the scale-invariant Morrey norm of curvature has strictly positive infimum on any nontrivial bundle. These statements extend classical gap theorems to a parabolic setting by using the energy-decreasing property of the flow.

Core claim

On an SU(r)-bundle of charge κ over the 4-sphere, the space of all connections with Yang-Mills energy less than 4 π² (|κ| + 2) deformation-retracts under Yang-Mills flow onto the space of instantons. On a compact quaternion-Kähler manifold with positive scalar curvature, the space of pseudo-holomorphic connections whose sp(1) curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, the infimum of the scale-invariant Morrey norm of curvature is positive.

What carries the argument

The Yang-Mills flow, which is energy-decreasing and provides the deformation retraction between the indicated connection spaces and the instantons.

If this is right

  • Taubes path-connectedness theorem for instanton moduli spaces follows from the retraction on the 4-sphere.
  • Pseudo-holomorphic connections with small sp(1) curvature component on quaternion-Kähler manifolds retract to instantons.
  • The scale-invariant Morrey norm of curvature is bounded below by a positive constant on any nontrivial bundle.
  • Classical analytic gap theorems admit parabolic strengthenings via the flow.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The retraction may be used to compute low-dimensional homotopy groups of the space of connections directly from the instanton component.
  • Similar energy thresholds could control bubbling analysis for sequences of connections on manifolds with positive Ricci curvature.
  • Numerical integration of the flow on explicit bundles over S^4 could test the sharpness of the constant 4π²(|κ| + 2).
  • The positivity of the Morrey-norm infimum implies that the only connection with vanishing Morrey norm on a nontrivial bundle is the zero connection, which is impossible.

Load-bearing premise

The manifold is either the 4-sphere or a compact quaternion-Kähler manifold with positive scalar curvature, and the connections satisfy the stated energy or Morrey-norm bound.

What would settle it

An explicit connection on an SU(r)-bundle over S^4 whose Yang-Mills energy lies strictly below 4π²(|κ| + 2) yet whose flow trajectory fails to converge to an instanton.

read the original abstract

We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an $\mathrm{SU}(r)$-bundle of charge $\kappa$ over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than $4 \pi^2 \left( |\kappa| + 2 \right)$ deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes's path-connectedness theorem. On a compact quaternion-K\"ahler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose $\mathfrak{sp}(1)$ curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves parabolic versions of gap theorems for the Yang-Mills energy. On an SU(r)-bundle of charge κ over S⁴, the space of connections with energy less than 4π²(|κ| + 2) deformation-retracts under the Yang-Mills flow onto the instanton space. On compact quaternion-Kähler manifolds with positive scalar curvature, pseudo-holomorphic connections with small Morrey norm of the sp(1) curvature component likewise deformation-retract onto instantons. On nontrivial bundles over compact manifolds of general dimension, the infimum of the scale-invariant Morrey norm of curvature is shown to be positive. These results are used to simplify Taubes's path-connectedness theorem.

Significance. If the proofs are complete, the parabolic deformation-retraction statements supply a dynamical approach to classical gap theorems, directly simplifying Taubes's theorem via the flow and extending gap phenomena to quaternion-Kähler settings and Morrey-norm controls. The results rest on standard well-posedness and ε-regularity properties of the Yang-Mills flow in the stated energy regimes, which are load-bearing but align with existing parabolic gauge-theory literature.

minor comments (3)
  1. [Introduction] Introduction: the statement that the S⁴ result 'simplifies the proof of Taubes's path-connectedness theorem' would be strengthened by a one-paragraph outline of the precise reduction (e.g., which homotopy is replaced by the flow).
  2. [Introduction] The abstract and introduction use the phrase 'parabolic versions of several known gap theorems' without citing the classical statements being parabolized; adding explicit references (e.g., to the original gap theorems) would improve readability.
  3. Notation for the charge κ and the constant 4π²(|κ| + 2) is introduced without a preliminary reminder of the normalization of the Yang-Mills energy functional; a short sentence recalling the convention would prevent minor confusion for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The referee's description of the main results is accurate. No specific major comments appear in the report, so we have no point-by-point responses to provide at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes parabolic gap theorems by invoking the standard well-posedness, energy monotonicity, and convergence properties of the Yang-Mills flow together with ε-regularity results already available in the gauge-theory literature. The central deformation-retract statements on SU(r)-bundles over S^4 and on quaternion-Kähler manifolds are obtained directly from these background facts once the energy or Morrey-norm threshold is imposed; no step is shown to reduce by the paper's own equations to a fitted parameter, a self-citation chain, or a renamed input. The simplification of Taubes's theorem is likewise a direct consequence of the new retraction and does not rely on any internal circular premise.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on classical background results in Yang-Mills theory and parabolic PDEs without introducing new free parameters or postulated entities.

axioms (2)
  • standard math The Yang-Mills flow exists globally or for sufficient time and decreases the energy functional on the space of connections
    Standard assumption invoked for the deformation-retraction statements.
  • domain assumption Instantons are the absolute minimizers of the Yang-Mills energy and their moduli spaces are well-understood
    Used to identify the target of the retraction.

pith-pipeline@v0.9.0 · 5686 in / 1312 out tokens · 35988 ms · 2026-05-23T06:41:15.190656+00:00 · methodology

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