Flowing maps to minimal surfaces: Existence and uniqueness of solutions

Rupflin, Melanie

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 349-368.

Nous étudions le nouveau flot géométrique introduit dans l'article [12] de Topping et de l'auteure, qui transforme un couple formé d'une application d'une surface vers une variété riemannienne et d'une métrique riemannienne du domaine. Ce flot change des données initiales appropriées en des immersions minimales ramifiées. Nous prouvons qu'une solution faible existe tant que le flot ne contracte pas une géodésique fermée du domaine en un point. De plus, nous montrons que cette solution est unique dans la classe des solutions faibles avec énergie décroissante. Ce travail complète l'article de Topping et de l'auteure [12] où le flot est introduit et où sa convergence asymptotique est étudiée.

We study the new geometric flow that was introduced in the paper [12] of Topping and the author that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper of Topping and the author [12] where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed.

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