Pseudodifferential structure, rigidity, and sharpness for the Steklov problem on differential forms

Abstract

We resolve four questions concerning Karpukhin's Dirichlet-to-Neumann operator on differential forms. First, for $0\leq p\leq n-2$, we prove that the operator is classical pseudodifferential of order one and compute its principal symbol, which is elliptic on the coclosed symbol space. This answers Open Question 7.19 in the survey of Colbois, Girouard, Gordon, and Sher [7]. Second, the symbol calculation yields a Weyl law with remainder $O(τ^{n-2})$, resolving the smooth case of their Open Question 7.23 in [7]. Third, in the critical dimension $n=2p+2$, we show that conformal deformations of the Euclidean ball fixing the boundary metric preserve the full operator. Nonisometric metrics therefore retain all equalities in Karpukhin's initial equality range, disproving the isometry-rigidity assertion of Conjecture 2.12 in [19](the rigidity assertion of Conjecture 7.27 in [7]). Fourth, we establish the all-index sharpness asserted in Conjecture 2.13 in [19](the sharpness assertion of Conjecture 7.27 in [7]) in the class of smooth compact connected oriented Riemannian manifolds with boundary. For $p\geq1$, a shrinking-core construction inspired by the collapsing models of Anné and Takahashi produces manifolds with fixed spherical boundary and explicit two-sided bounds for an initial eigenvalue cluster.