Geometric renormalization below the ground state (Q2909672)

This paper deals with the harmonic map heat flow, posed on \(\mathbb R^+_s\times\mathbb R^2_x\), with a general target manifold. The main results areNEWLINENEWLINE\(\bullet\) a global existence, uniqueness and smoothness result for the solution \( \varphi \) to the heat flow associated to classical initial data \( \varphi_0 \) (ie, in the extrinsic formulation, a Schwartz function up to an additive constant) with energy below the critical threshold.NEWLINENEWLINE\( \bullet \) For a given boundary frame \( e (\infty) \), that is an orthonormal frame at \(\varphi (x=\infty)\); there exists a unique caloric gauge associated to \(\varphi\).NEWLINENEWLINE \(\bullet\) This construction can be adapted to the dynamic setting, that is when the initial data depends on a time parameter \(t\) (and is, uniformly in \(t\), below the energy threshold).