Higher-dimensional Thorpe correction for action operators

Establish that, for every Euclidean vector space $V$ and every differential form $\omega\in\Lambda^pV^*$, there exists a $4$-form $\eta\in\Lambda^4V^*$ such that the corrected action operator satisfies $0\leq\mathcal A_\omega+\widehat\eta\leq 2|\omega|^2\operatorname{id}_{\Lambda^2V}$, thereby extending the six-dimensional three-form result to higher dimensions.

Background

The paper introduces a Thorpe-type modification of the action operator Aω\mathcal A_\omega by adding an operator η^\widehat\eta induced by a $4$-form. This modification does not change the curvature term in the Lichnerowicz Laplacian because algebraic curvature operators are orthogonal to operators induced by $4$-forms.

For $3$-forms in dimension six, Theorem \ref{ThorpeTrick36} proves the bound 0≤Aω+η^≤2∣ω∣2id⁡0\leq\mathcal A_\omega+\widehat\eta\leq2|\omega|^2\operatorname{id}. The conjecture asks whether the same optimal bound holds for arbitrary form degree and dimension. Example \ref{ExampleSharpCandidate} shows that the constant $2$ cannot be improved.

References

More generally, we conjecture that an analogous theorem holds in higher dimensions. Specifically, we ask for \begin{conjecture} \label{ConjectureControllingLichnerowicz} Let $(V,g)$ be a Euclidean vector space. For every $\omega \in p V{*}$ there exists $\eta \in 4V*$ such that \begin{align*} 0 \leq \mathcal{A}{\omega} + \hat{\eta} \leq 2 |\omega|2 \id{2V}. \end{align*} \end{conjecture}

— A Thorpe Trick for the Bochner Technique  (2609.11653 - Wink, 10 Sep 2026) in Conjecture \ref{ConjectureControllingLichnerowicz} in the Introduction