Local rigidity of conformally Euclidean metrics for the anisotropic Helmholtz equation at fixed energy

We prove local rigidity of smooth conformally Euclidean metrics $g_0$ on smooth compact domains $M\subset {\mathbb{R}}^n$, $n\ge 3$, for the anisotropic Helmholtz equation $(-{\mathrm{\Delta }}_g-k^2)u=0$ at a fixed energy $k^2>0$ which is not a Dirichlet eigenvalue of $-{\mathrm{\Delta }}_{g_0}$ in $M$. Any smooth metric sufficiently close to $g_0$ in $H^s(M)$, with integer $s>n\mathrm{/}2+1$, and having the same Dirichlet-to-Neumann map, is its pullback by a smooth diffeomorphism fixing the boundary pointwise. The proof uses a Helmholtz embedding to fix the gauge and a compactness argument together with a characterization of the kernel of the linearization.