Failure of strong unique continuation for harmonic functions on RCD Spaces

Unique continuation of harmonic functions on $RCD$ space is a long-standing open problem, with little known even in the setting of Alexandrov spaces. In this paper, we establish the weak unique continuation theorem for harmonic functions on $RCD(K,2)$ spaces and give a counterexample for strong unique continuation in the setting of $RCD(K,N)$ space for any $N\geq 4$ and any $K\in \mathbb{R}$.