Corrigenda to “Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation”

Abstract:

The proof of Lemma 4.4 in our article, which appeared in Trans. Amer. Math. Soc. 371 (2019), 8781–8810, contains a flaw. In proving the existence of a minimizer of the map $\mathbf {A} \mapsto I_\epsilon [\mathbf {A}]$ defined therein, we stated that this map is a convex function of $\mathbf {A}$. This is incorrect, as $I_\epsilon$ is a composition of two convex functions, a quadratic form and an absolute value, and since the absolute value function is not monotonic, there is no guarantee that the resulting functional is convex. This short article corrects this flaw by showing that there is a continuous convex functional $J_\epsilon$ such that $I_\epsilon [\mathbf {A}] = J_\epsilon [\mathbf {A}^2]$, and then employing weak lower semi-continuity of $J_\epsilon$ to demonstrate the existence of a minimizer of $I_\epsilon$.