Hyperbolic nonlinear Schr\"odinger equations on $R× T$

In this paper, we consider the hyperbolic nonlinear Schr\"odinger equations (HNLS) on R. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for higher odd nonlinearities. Moreover, when the initial data is small, we prove the global existence and scattering for the solutions to HNLS with higher nonlinearities (except the cubic one) in critical Sobolev spaces. The main ingredient of the proof is the sharp up to the endpoint local/global-in-time Strichartz estimates.