Geometric renormalization below the ground state

TL;DR

This paper extends the caloric gauge framework to a bounded geometry setting for Schrödinger maps, reaching up to the ground state energy, advancing the understanding of geometric renormalization in critical wave maps.

Contribution

It develops a caloric gauge construction applicable up to the ground state energy in a bounded geometry setting for Schrödinger maps.

Findings

Caloric gauge construction valid up to the ground state energy.

Extension of geometric renormalization techniques to bounded geometry.

Improved understanding of Schrödinger maps in critical regimes.

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 Schr\"odinger 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.