Proof of geometric Borg's Theorem in arbitrary dimensions

Let $\Delta+V$ be the discrete Schr\"odinger operator, where $\Delta$ is the discrete Laplacian on $\mathbb{Z}^d$ and potential $V:\mathbb{Z}^d\to \mathbb{C}$ is $\Gamma$-periodic with $\Gamma=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$. In this study, we establish a comprehensive characterization of complex-valued $\Gamma$-periodic functions such that the Bloch variety of $\Delta+V$ contains a graph of an entire function, in particular, we show that there are exactly $q_1q_2\cdots q_d$ such functions (up to Floquet isospectrality and translation). Moreover, by applying this understanding to real-valued functions $V$, we prove that $V$ is constant if and only if the Bloch variety of $\Delta+V$ contains a graph of an entire function, which confirms the conjecture concerning the geometric version of Borg's theorem in arbitrary dimensions.