On non-homeomorphic surfaces with close DN maps

Let $(M,g)$ be a genus $m$ surface with boundary $\mathrm{\Gamma }$ and DN map $\mathrm{\Lambda }$. Introduce the Schottky double $2M$ of $(M,g)$ and denote by $Sys(2M)$ the length of the shortest closed geodesics in the hyperbolic metrics on $2M$. We prove that $Sys(2M)$ is small if $\mathrm{\Lambda }$ is close, in the operator norm, to the DN map ${\mathrm{\Lambda }}_{\ast }$ of some surface $(M_{\ast },g_{\ast })$ of lower genus $m_{\ast }<m$ with the same boundary $\mathrm{\Gamma }$: $$\mathrm{\parallel }\mathrm{\Lambda }-{\mathrm{\Lambda }}_{\ast }{\mathrm{\parallel }}_{B(H^{1\mathrm{/}2}(\mathrm{\Gamma });H^{-1\mathrm{/}2}(\mathrm{\Gamma }))}\to 0⟹Sys(2M)\to 0.$$