Deformation of singular connections I: $G_2-$instantons with point singularities

Abstract

In dimension 7, we establish a Fredholm theory for a Dirac-type operator associated to a connection with point singularities. There are two applications. $1$. over a closed 7-manifold, under some natural conditions, a $G_2-$instanton and its point singularities can still be "seen" when the $G_2-$structure is properly perturbed. $2$. over a ball in ${\mathbb{R}}^7$, for any almost-Euclidean $G_2-$structure, there exists a $G_2-$monopole asymptotic to an arbitrary Hermitian Yang-Mills connection on $S^6$.

60 pages. Minor change from v1: References added and mentioned. Definition 2.14 added. Some change in page 41 on the proof of Thm 1.13. Two typos are corrected. Nothing else is touched

References in corpus (1)

• The Yamabe problem on stratified spaces

Cited by in corpus (2)

• An elliptic theory of indicial weights and applications to non-linear geometry problems
• Local solution to the $G_2-$monopole equation with prescribed tangent cone and $G_2-$structure