Jollivet–Sharafutdinov inverse Steklov spectral conjecture

Let $\Omega$ be a smooth surface with boundary $M=\partial\Omega$, and let $g_1$ and $g_2$ be Riemannian metrics on $\Omega$. Two metrics are $\sigma$-isometric when there exist a diffeomorphism $\Phi:\Omega\to\Omega$ and a function $\rho\in C^\infty(\Omega;\mathbb{R}_+)$ such that $\rho|_M=1$ and

$$g_2=\rho\Phi^*g_1.$$

Jollivet–Sharafutdinov inverse Steklov spectral conjecture. The metrics $g_1$ and $g_2$ have the same Steklov spectrum if and only if they are $\sigma$-isometric. This conjecture identifies the natural obstruction to recovering a surface metric from its Steklov spectrum: boundary-fixing conformal changes preserve the Dirichlet-to-Neumann operator, while the conjecture asserts that these changes and diffeomorphisms account for all isospectrality. Its status is not resolved in the supplied text.