Rigidity of a trace estimate for Steklov eigenvalues (Q2223600)

The authors consider the Steklov eigenvalue problem \[ \left\{ \begin{aligned} \Delta u &= 0\\ \frac{\partial u}{\partial \nu} &= \sigma u \end{aligned} \right. \] on a compact \(n\)-dimensional Riemannian manifold \((M,g)\) with boundary. In a previous work [\textit{Y. Shi} and \textit{C. Yu}, J. Differ. Equations 261, No. 3, 2026--2040 (2016; Zbl 1337.35101)] they had proved that the first \(m\) nonzero Steklov eigenvalues \((0 = \sigma_0 < ) \sigma_1 \leq \sigma_2 \leq \ldots \leq \sigma_m\) satisfy the bound \[\sigma_1 + \sigma_2 + \ldots + \sigma_m \leq \frac{\textrm{Vol}\,(\partial M)}{\textrm{Vol}\,(M)},\tag{1} \] where \(m > 0\) is the dimension of the space of parallel exact 1-forms on \(M\). In the paper under review the authors characterize the case of equality in (1): their main theorem (Theorem 1.2) states that there is equality if and only if \(M\) is a direct product of the ball \(\mathbb{B}^m (R)\) of radius \(R = \frac{m\textrm{Vol}\,(\partial M)}{\textrm{Vol}\,(M)}\), and a closed manifold \(F\) which is required to satisfy a lower bound on its first positive Laplacian eigenvalue, which also involves \(R\). The proof is based on a new triviality result for certain Riemannian submersions between complete manifolds with boundary.