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The caloric gauge was introduced in [38] by Tao with studying large data energy-critical wave maps mapping from  R  2+1  to hyperbolic space  H   m   in view. In [1] Bejenaru et al. adapted the caloric gauge to the setting of Schrödinger maps from  R   d +1  to the standard sphere  S  2 ↪ R  3  with initial data small in the critical Sobolev norm. Here we develop the caloric gauge in a bounded geometry setting with a construction valid up to the ground-state energy, which for mapsis the natural limitation imposed by the harmonic map heat flow used to define the caloric gauge. In [34] we apply these results to the study of Schrödinger maps lying below the ground-state energy.

Geometric Renormalization Below the Ground State