We have studied the iso-height lines on the $mathrm{WO_3}$ surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical Ising model
. They belong to the family of conformal invariant curves called Schramm-Loewner evolution (or $SLE_{kappa}$), with diffusivity of $kappa sim 3$. This can be regarded as the first experimental observation of SLE curves. We have also argued that Ballistic Deposition (BD) can serve as a growth model giving rise to contours with similar statistics at large scales.
