Cubic NLS on 2d manifolds

Cubic NLS on RxT and T2

• Scaling is $s_{c}\,$= 0.
• For $R\times T$ one has LWP for $s>0\,$ TkTz-p2.
• For $T\times T$ one has LWP for $s>0\,$ Bo1993.
• In the defocussing case one has GWP for $s>1\,$ in both cases by Hamiltonian conservation.
  • On $T\times T$ one can improve this to $s>2/3\,$ by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).
• In the focusing case on $T\times T$ one has blowup for data close to the ground state, with a blowup rate of $(T^{*}-t)^{-1}\,$ BuGdTz-p

Cubic NLS on the sphere S^2

• If instead one considers the sphere $S^{2}\,$ then uniform local well-posedness fails for $3/20<s<1/4\,$ BuGdTz2002, Ban-p, but holds for $s>1/4\,$ BuGdTz-p7.
  • For $s>1/2\,$ this is in BuGdTz-p3.
  • These results for the sphere can mostly be generalized to other Zoll manifolds.