Local Uniform Well-posedness for Nonlinear Schrödinger Equation with General Nonlinearity on Product Manifolds

We study the Cauchy problem of the nonlinear Schr & ouml;dinger equation posed on a (d + 1)-dimensional product manifolds M=ZdxS1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M={Z<^>{d}} \times {{\mathbb S}<^>1}$$\end{document}, where Zd denotes the d-dimensional compact manifold under the eigenvalue assumption. We prove the uniform local well-posedness of Hs solution with the defocusing subcritical nonlinearity and extend the results of [Amer. J. Math., 2004, 126(3): 569-605], [Ann. Sci. & Eacute;cole Norm. Sup. (4), 2005, 38(2): 255-301] and [Sci. China Math., 2015, 58(5): 1023-1046] to the more general geometry. The main tools in our argument are multilinear estimates and Bony's linearized technique.