arXiv subjects

Enric Solé-Farré

Publications and source records attributed to Enric Solé-Farré.

3 recordsLinked to original sources

arXiv2026

Gauge theory on Cohomogeneity One Spin(7)-Manifolds from Aloff--Wallach Spaces

We begin a systematic study of $\mathrm{Spin}(7)$-instantons on complete non-compact cohomogeneity one $\mathrm{Spin}(7)$-manifolds with Aloff--Wallach spaces $N_{k,l}=\mathrm{SU}(3)/\mathrm{U}(1)_{k,l}$ as principal orbits. We provide a complete classification of admissible principal $\mathrm{U}(1)$- and $\mathrm{SO}(3)$-bundles and smooth bounded solutions to the $\mathrm{Spin}(7)$-instanton equation on them. We study bubbling phenomena for the constructed $\mathrm{Spin}(7)$-instantons and discuss the compactification of the moduli space of $\mathrm{Spin}(7)$-instantons in both the ALC and AC settings.

math.DG↗

arXiv2024

The Hitchin index in cohomogeneity one nearly Kähler structures

Nearly Kähler and Einstein structures admit a variational characterization, where the second variation is associated with a strongly elliptic operator. This allows us to associate a Morse-like index to each structure. Our study focuses on how these indices behave under the assumption that the nearly Kähler structure admits a cohomogeneity one action. Specifically, we investigate elements of the index that also exhibit cohomogeneity one symmetry, reducing the analysis to an ODE eigenvalue problem. We apply our discussion to the two inhomogeneous examples constructed by Foscolo and Haskins. We obtain non-trivial lower bounds on the Hitchin index and Einstein co-index for the inhomogeneous nearly Kähler structure on $S^3\times S^3$, answering a question of Karigiannis and Lotay.

math.DG↗

arXiv2024

Stability of nearly Kähler and nearly parallel $G_2$-manifolds

We investigate generalisations of Hitchin's functionals, whose critical points correspond to nearly Kähler and nearly parallel $G_2$-structures. Our focus is on the gradient flow of these functionals and the spectral decomposition of their Hessians with respect to natural indefinite inner products. We introduce a Morse-like index for these functionals, termed the Hitchin index. We prove this index provides a lower bound for the Einstein co-index and explore its relationship with the deformation theory of $G_2$- and $\operatorname{Spin}(7)$-conifolds.

math.DG↗