Calderon inverse Problem with partial data on Riemann Surfaces

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schr"odinger operator $\mathrm{\Delta }+V$ with potential $V\in C^{1,\alpha}(M_0)$forsome$\alpha>0$,theDirichlet-to-Neumannmap$N|_{\Gamma}$measuredonanopenset$\Gamma\subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.