Sharp $L^4$ Strichartz estimate for Hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$

We prove the sharp $L^4$ Strichartz estimate without derivative loss for the hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$, \begin{equation} |e^{it (\partial_{x_{1}}^2-\partial_{x_{2}}^2)} φ|{L^4{t,x_{1},x_{2}}([0,1]\times $\mathbb{R}$ \times $\mathbb{T}$)}\lesssim |φ|{L{x_{1},x_{2}}^2($\mathbb{R}$ \times $\mathbb{T}$)}, \end{equation} which serves as the hyperbolic analogue of the classical result of Takaoka-Tzvetkov \cite{takaoka20012d}. The proof is based on the combination of a robust kernel decomposition method with precise measure estimates for semi-algebraic sets. As an immediate application, we establish the global well-posedness for the cubic hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$ in the $L^2$-critical space with sufficiently small initial data.