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SUMMARY:Jesse Madnick (National Center for Theoretical Sciences)
DTSTART:20210505T090000Z
DTEND:20210505T100000Z
DTSTAMP:20260423T021402Z
UID:VSGS/27
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/VSGS/27/">Th
 e Second Variation of Holomorphic Curves in the 6-Sphere</a>\nby Jesse Mad
 nick (National Center for Theoretical Sciences) as part of Virtual seminar
  on geometry with symmetries\n\n\nAbstract\nThe 6-sphere is the only $n$-s
 phere with $n > 2$ that admits an almost-complex structure.  Equipping the
  round 6-sphere with its standard ($G_2$-invariant) almost-complex structu
 re\, the holomorphic curves in $S^6$ are minimal surfaces\, and play an im
 portant role in $G_2$-geometry.  These surfaces exist in abundance: by a r
 emarkable theorem of Bryant\, extended by Rowland\, every closed Riemann s
 urface may be conformally embedded in $S^6$ as a holomorphic curve of "nul
 l-torsion."\n\nWhile holomorphic curves in $S^6$ are area-minimizing to fi
 rst order\, they are not area-minimizing to second order.  This failure is
  encoded by the spectrum of the Jacobi operator\, which contains informati
 on such as the Morse index and nullity.  For closed\, null-torsion holomor
 phic curves of low genus\, we explicitly compute the multiplicity of the f
 irst Jacobi eigenvalue.  Moreover\, for all genera\, we give a simple lowe
 r bound for the nullity in terms of the area and genus.  Time permitting\,
  we will also outline some recent results in the setting of holomorphic cu
 rves with boundary.\n
LOCATION:https://researchseminars.org/talk/VSGS/27/
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