Eigenvalue Estimate for the Rough Laplacian on $1$-Forms and its Applications

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ 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 $\lambda^{(1)}_{1}$ 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.