Harmonic 3-spheres, or normalized harmonic map heat flow
speaker: Michael Struwe (ETH, Zürich)abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the direct method fails, as the infimum of Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow.
To overcome these difficulties, we propose the normalized harmonic map heat flow as a new tool, and we show that for this flow the identity map from the 3-sphere to itself now, indeed, is stable; moreover, the flow converges to a harmonic 3-sphere also when we perturb the target geometry. While our results are strongest in the perturbative setting, we also outline a possible global theory, which may open up a rich research agenda.
timetable:
Fri 18 May, 11:40 - 12:30, Aula Dini
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