Eigenvalue and gap estimates of isometric immersions for the Dirichlet-to-Neumann operator acting on \(p\)-forms
Let \((M,g)\) be an \((n+1)\)-dimensional compact oriented Riemannian manifold with smooth boundary \(\partial M\) isometrically immersed in some Euclidean space. By using the extension of the Dirichlet-to-Neumann operator (also called Steklov operator) \(T^{[p]}\) acting on the bundle of \(p\)-forms \(\bigwedge^p(\partial M)\) developed in [\textit{S. Raulot} and \textit{A. Savo}, J. Funct. Anal. 262, No. 3, 889--914 (2012; Zbl 1250.58013)], the article under review gives a lower bound for \(\nu_{1,p}-\nu_{1,p-1}\), where \(\nu_{k,p}\) denotes the \(k\)-th eigenvalue of \(T^{[p]}\), in terms of the curvature of \(M\). The proof follows the method developed in [\textit{P. Guerini} and \textit{A. Savo}, Trans. Am. Math. Soc. 356, No. 1, 319--344 (2004; Zbl 1034.58026)], where the Hodge-Laplace operator acting on \(p\)-forms was considered.
scientific article; zbMATH DE number 7034594