Some sharp Hodge Laplacian and Steklov eigenvalue estimates for differential forms (Q283566)

The main result of this paper is the following: Theorem 1. Let \((N^n,g)\) be a compact oriented Riemannian manifold \((n\geq 2)\) with smooth boundary \(\Sigma\). Suppose the Bochner curvature \(W^r\) or \(W^{n-r}\) is bounded from below by \(k\geq 0\). Assume that the \(q\) curvature \(s_q\) of \(\Sigma\) is nonnegative, where \(q=\min\{r,n-r\}\). Then for \(1\leq r\leq n-1\), we have \[ 2\lambda'_{1,r}=2\lambda''_{1,r-1}\geq k+s_rs_{n-r}+\sqrt{(s_rs_{n-r})^2+2s_rs_{n-r}k}, \] where \(\lambda'_{1,r}\) (resp. \(\lambda''_{1,r}\)) is the first nonzero eigenvalue of the Hodge Laplacian of the exact (resp. co-exact) \(r\) forms on \(\Sigma\). This theorem provides sharp lower bounds for the first nonzero Hodge-Laplacian eigenvalue and extends the results of \textit{C. Xia} [Proc. Am. Math. Soc. 125, No. 6, 1801--1806 (1997; Zbl 0882.53028)] and \textit{S. Raulot} and \textit{A. Savo} [J. Geom. Anal. 21, No. 3, 620--640 (2011; Zbl 1228.58018)]. Similar results for the first nonzero Steklov eigenvalue for differential forms are also given.