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In this paper, we establish bilinear and gradient bilinear Strichartzestimates for Schr\"odinger operators in 2 dimensional compact manifolds withboundary. Using these estimates, we can infer the local well-posedness of cubicnonlinear Schr\"odinger equation in $H^s$ for every $s>{2/3}$ on suchmanifolds.

Bilinear Strichartz estimates for Schr\"odinger operators in 2 dimensional compact manifolds with boundary and cubic NLS