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BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Circle-valued maps: bubbles and singularities Mironescu, Petru


We consider maps u : Ω → S1 having some Sobolev regu- larity u ∈ Ws,p. For values of s and p relevant for applications, a) either such maps need not have a phase φ with the same regularity as u b) or the phase φ exists but is not controlled by the norm of u. In case a), “factorization allows to write each such u as u = v w, where v lifts and w is “smoother than u. In the first part of the talk, I will present a new, very simple proof of this result, based on the theory of weighted Sobolev spaces. Case b) occurs only in dimension one. In that case, there is an explicit example of loss of control (a “kink), and it turns out that only kinks lead to loss of control. This translates into a description of weakly convergent sequences which complements the one given by the theory of Cartesian currents. A quantitative result concerning kinks leads to the existence of minimal maps winding once around the unit circle. The common theme of the proofs of the above results is the geometric detection of the energy concentration of manifold-valued maps.

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