| The caloric gauge was introduced in [Tao04] by Tao with studying large data energy-critical wave maps mapping from R2+1 to hyperbolic space Hm in view. In [BIKT08], Bejenaru, Ionescu, Kenig, and Tataru adapted the caloric gauge to the setting of Schrodinger maps from R d+1 to the standard sphere S 2 ↪ R3 with initial data small in the critical Sobolev norm. Chapters 1--5 develop the caloric gauge in a bounded geometry setting with a construction valid up to the ground state energy, which for maps R2+1 → Mm is the natural limitation imposed by the harmonic map heat flow used to define the caloric gauge. This construction finds application in Chapters 6--12, which consider the Schrodinger map initial value problem 6t4=4x D44x ,0=40 x, with ϕ0: R2 → S2 ↪ R3 a smooth HinfinityQ map from the Euclidean space R2 to the sphere S2. The caloric gauge is but one tool used in proving the main result for Schrodinger maps, which may be interpreted as a step toward verifying the threshold conjecture. In particular, it is shown that, given energy-dispersed data ϕ0 with subthreshold energy, the Schrodinger map system then admits a unique global smooth solution ϕ ∈ C(R → HinfinityQ ) provided that the gradient ∂xϕ respects an a priori L4 boundedness condition. Also shown are global-in-time bounds on certain Sobolev norms of ϕ. Toward this end, improved local smoothing and bilinear estimates are shown via an adaptation of the Planchon-Vega approach to such estimates to the nonlinear (or linear covariant) setting of Schrodinger maps. |
