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SUMMARY:Jesse Madnick
DTSTART:20210715T230000Z
DTEND:20210716T000000Z
DTSTAMP:20260418T130018Z
UID:2ndGAF/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/2ndGAF/5/">T
 he Jacobi Spectrum of Null-Torsion Holomorphic Curves in the 6-Sphere</a>\
 nby Jesse Madnick as part of The 2nd Geometric Analysis Festival\n\n\nAbst
 ract\nMinimal surfaces are area-minimizing to first order\, but not necess
 arily to second-order.  The extent to which a minimal surface is (or isn't
 ) area-minimizing to second-order is encoded by its Jacobi operator. Howev
 er\, for a given minimal surface\, computing the spectrum of the Jacobi op
 erator — i.e.\, the eigenvalues and their multiplicities — is generall
 y a non-trivial task. \n\\indent In this talk\, we will discuss a class of
  minimal surfaces in the round 6-sphere known as “null-torsion holomorph
 ic curves.”  These surfaces are of interest to $G_2$ geometry\, and exis
 t in abundance.  Indeed\, by a remarkable theorem of Bryant\, extended by 
 Rowland\, every closed Riemann surface may be conformally embedded as a nu
 ll-torsion holomorphic curve in $S^6$. \n\\indent For null-torsion holomor
 phic curves of low genus\, we will compute the multiplicity of the first J
 acobi eigenvalue.  Moreover\, for all genera\, we will give a simple lower
  bound for the nullity (the multiplicity of the zero eigenspace) in terms 
 of the area and genus.  We expect that these results will have implication
 s for the deformation theory of asymptotically conical associative 3-folds
  in euclidean $R^7$.\n
LOCATION:https://researchseminars.org/talk/2ndGAF/5/
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