Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications
Teng Huang, Weiwei Wang
TL;DR
The paper investigates geometric lower bounds for the first positive eigenvalue of the rough Laplacian on 1-forms on closed 2n-dimensional manifolds with nonzero Euler characteristic, showing that Li-Yau-type bounds do not hold universally for 1-forms. It develops a framework combining spectral theory, Poincaré–Sobolev inequalities under curvature and diameter bounds, and L^{2p}-norm control of the Riemann tensor to derive explicit lower bounds for λ1^{(1)}. A key non-vanishing criterion links small eigenvalues to nowhere-vanishing eigenforms, which, together with topology, yields vanishing Euler characteristic under certain curvature conditions. The results further connect spectral data to symmetry via Killing fields, establishing finiteness results for the isometry group and extending Bochner-type rigidity theorems to settings with integral curvature bounds and Ricci constraints.
Abstract
In this article, we establish a geometric lower bound for the first positive eigenvalue $λ^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic. In contrast to the case of functions, such a Li-Yau-type estimate does not hold in general, as evidenced by existing counterexamples. Under assumptions including a lower bound on Ricci curvature, an upper bound on diameter, and an $L^{2p}$-norm bound on the Riemann curvature tensor, we prove that $λ^{(1)}_{1}$ is bounded below by a positive constant depending on these parameters. As applications, we derive vanishing results for the Euler characteristic under certain Ricci curvature bounds and the presence of a nonzero Killing vector field, extending classical Bochner-type theorems.