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Geometry-Topology Seminar

Tuesday, November 13, 2018 - 4:30pm

Rob Kusner

UMass

Location

University of Pennsylvania

DRL 4N30

Minimal surfaces in the n-sphere are critical for the
Willmore energy W = ∫(1 + H2)dA.  Their Willmore stability is
equivalent to a spectral gap in the Jacobi operator for area between
the eigenvalues -2 and 0.  The square Clifford 2-torus in the 3-sphere
has this spectral gap, as does the equilateral minimal 2-torus in the
5-sphere, so both are Willmore stable.  They're also the only minimal
2-tori embedded by first eigenfunctions of the Laplacian, and this
implies they remain Willmore stable in all n-spheres as well (in fact,
for any minimal surface in the n-sphere, this "persistent Willmore
stability" is equivalent to being embedded by first eigenfunctions),
but the equilateral torus has nontrivial 3rd variation of W, and thus
is not a W-minimizer (though if we fix the conformal type, it is)!
While this gives some evidence that Willmore Conjecture holds in every
codimension, further support is our area-index characterization of the
Clifford torus.  Finally, for the higher genus minimal surfaces
(e.g. those constructed by Lawson and those by
Karcher–Pinkall–Sterling) in the 3-sphere which Choe–Soret showed are
embedded by first eigenfunctions, we discovered that they are (up to
Möbius transformations of the n-sphere) the unique W-minimizers in
their conformal class.  This "conformal rigidity" result has a
striking intrinsic corollary — various minimal surfaces of the same
genus must be distinct Riemann surfaces — and is another bit of
evidence for the conjecture that Lawson's simplest surfaces are the
W-minimizers among all surfaces of fixed genus.