BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Henrique N. Sá Earp (University of Campinas (Unicamp)) DTSTART:20210728T160000Z DTEND:20210728T170000Z DTSTAMP:20260423T021219Z UID:VSGS/33 DESCRIPTION:Title: Ha rmonic $\\rm{Sp}(2)$-invariant $\\rm{G}_2$-structures on the $7$-sphere\nby Henrique N. Sá Earp (University of Campinas (Unicamp)) as part of V irtual seminar on geometry with symmetries\n\n\nAbstract\nWe describe the $10$-dimensional space of $\\rm{Sp}(2)$-invariant $\\rm{G}_2$-structures o n the homogeneous $7$-sphere $\\mathbb{S}^7=\\mathrm{Sp}(2)/\\rm{Sp}(1)$ a s $\\Omega_+^3(\\mathbb{S}^7)^{\\mathrm{Sp}(2)}\\simeq \\mathbb{R}^+ \\tim es\\rm{Gl}^+(3\,\\mathbb{R})$. \n In those terms\, we formulate a gener al Ansatz for $\\rm{G}_2$-structures\, which realises representatives in e ach of the $7$ possible isometric classes of homogeneous $\\rm{G}_2$-struc tures.\n Moreover\, the well-known nearly parallel ${round}$ and ${squ ashed}$ metrics occur naturally as opposite poles in an $\\mathbb{S}^3$-fa mily\, the equator of which is a new $\\mathbb{S}^2$-family of coclosed $ \\rm{G}_2$-structures satisfying the harmonicity condition $\\mathrm{div}\ \\; T=0$. \n We show general existence of harmonic representatives of $ \\rm{G}_2$-structures in each isometric class through explicit solutions o f the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points\, showing explicit examples of degenerate and nondegenerate local m axima and minima\, at various regimes of the general Ansatz. Finally\, for metrics outside of the Ansatz\, we identify families of harmonic $\\rm{G} _2$-structures\, prove long-time existence of the flow and study the stabi lity properties of some well-chosen examples.\n\nJoint work with E. Loubea u\, A. Moreno and J. Saavedra.\n LOCATION:https://researchseminars.org/talk/VSGS/33/ END:VEVENT END:VCALENDAR