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On Schrödinger maps from \(T^1\) to \(S^2\). (À propos des Schrödinger maps de \(T^1\) dans \(S^2\).) (English. French summary) Zbl 1308.58023
Summary: We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from \(T^1\) to \(S^2\) This estimate yields some continuity properties of the flow map for the topology of \(L^2(T^1,S^2)\), provided one takes its quotient by the continuous group action of \(T^1\) given by translations. We also prove that without taking this quotient, for any \(t>0\) the flow map at time \(t\) is discontinuous as a map from \(\mathcal C(T^1,S^2)\), equipped with the weak topology of \(H^{1/2}\) to the space of distributions \((\mathcal C(T^1,S^2))^*\). The argument relies in an essential way on the link between the Schrödinger map equation and the binormal curvature flow for curves in the euclidean space, and on a new estimate for the latter.

MSC:
58J99 Partial differential equations on manifolds; differential operators
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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