English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The $\theta$-Kapustin-Witten equations are a family of equations for aconnection $A$ on a principal $G$-bundle $E \to W^4$ and a one-form $\phi$,called the Higgs field, with values in the adjoint bundle $\operatorname{ad}E$. They give rise to second-order partial differential equations that can bestudied more generally on Riemannian manifolds $W^n$ of dimension $n$. For$G=SU(2)$, we report a dichotomy that is satisfied by solutions of thesecond-order equations on Ricci-flat ALX spaces with sectional curvaturebounded from below. This dichotomy was originally established by Taubes for$W^n=\mathbb{R}^n$; the alternatives are: either the asymptotic growth of theaveraged norm of the Higgs field over geodesic spheres is larger than apositive power of the radius, or the commutator $[\phi\wedge\phi]$ vanisheseverywhere. As a consequence, we are able to confirm a conjecture by Nagy andOliveira, namely, that finite energy solutions of the $\theta$-Kapustin-Wittenequations on ALE and ALF gravitational instantons with $\theta\neq 0$ are suchthat $[\phi\wedge\phi]=0$, $\nabla^A \phi=0$, and $A$ is flat.

Growth of the Higgs Field for Kapustin-Witten solutions on ALE and ALF gravitational instantons