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Isospectral finiteness on hyperbolic 3-manifolds
Interscience Publishers 2006 Communications on pure and applied mathematics Vol.59 No.5
<P>In this paper we show that for a given set of lengths of closed geodesics, there are only finitely many convex-cocompact, hyperbolic 3-manifolds with incompressible boundary, up to orientation-preserving isometries. © 2005 Wiley Periodicals, Inc.</P>
Schrödinger flow near harmonic maps
Gustafson, Stephen,Kang, Kyungkeun,Tsai, Tai-Peng Interscience Publishers 2007 Communications on pure and applied mathematics Vol.60 No.4
<P>For the Schrödinger flow from ℝ<SUP>2</SUP> × ℝ<SUP>+</SUP> to the 2-sphere 𝕊<SUP>2</SUP>, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blowup time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to 0. © 2006 Wiley Periodicals, Inc.</P>