Geometric Analysis and Topology Seminar

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Morse index stability for bubbling in dimension two

Speaker: Matilde Gianocca, ETH

Location: Warren Weaver Hall 512

Date: Friday, April 18, 2025, 11 a.m.

Synopsis:

In this talk, I will discuss compactness properties of sequences of (approximate) harmonic maps in two dimensions, with a focus on their energy distribution and stability. A well-known result in this context is the energy identity, which asserts that the total energy of a sequence of harmonic maps converges to the sum of the energy of the weak limit and the energies of finitely many harmonic "bubbles". The limiting objects (unions of harmonic maps with some additional properties) are called "bubble trees".
Building on this, I will present a joint result with T. Rivière and F. Da Lio, showing that the extended Morse index, a measure of instability for the given harmonic map, is upper semi-continuous in the bubble tree convergence.
The main ideas and techniques involved in the proof will be outlined. If time permits, I will also discuss how these methods extend to Ginzburg-Landau sequences, which approximate harmonic maps into spheres (joint with F. Da Lio).