Geometric renormalization below the ground state

Abstract

The caloric gauge was introduced by Tao with studying large data energy critical wave maps mapping from ${\mathbf{R}}^{2+1}$ to hyperbolic space ${\mathbf{H}}^m$ in view. In \cite{BIKT} Bejenaru, Ionescu, Kenig, and Tataru adapted the caloric gauge to the setting of Schrödinger maps from ${\mathbf{R}}^{d+1}$ to the standard sphere $S^2\hookrightarrow {\mathbf{R}}^3$ 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.

39 pages; Typos and argument for noncompact target manifolds corrected; Published form available at http://imrn.oxfordjournals.org/cgi/content/abstract/rnr169?ijkey=OXVb8Exb1XuXqLz&keytype=ref

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• Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions
• Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces
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• Asymptotic behaviors of Landau-Lifshitz flows from ${\mathbb{R}}^2$ to Kähler manifolds
• Bilinear Strichartz estimates for the Schr{ö}dinger map problem
• An unconstrained Lagrangian formulation and conservation laws for the Schrödinger map system
• Global well-posedness for the Schrödinger map problem with small Besov norm