Vortragsveranstaltung
On the Inner Radius of Nodal Domains
Dan Mangoubi
12.2.2008, 15:30 Uhr
Let M be a closed Riemannian manifold of dimension n.
Let f be an eigenfunction of the Laplacian on M with eigenvalue k.
A nodal domain is a connected component of the set f <> 0.
We discuss the asymptotic geometry of nodal domains on M.
We prove that the inner radius R of a nodal domain is bounded by
C_1 / k > R > C_2 / ^(n-1)/2 .
In dimension two we have a sharp bound.
One ingredient of our proof is the estimation of the volume
of positivity of a harmonic function u in the unit ball with u(0)=0,
in terms of its growth.
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