Spectral asymptotics for kinetic Brownian motion on Riemannian manifolds

We prove the convergence of the spectrum of the generator of the kinetic Brownian motion to the spectrum of the base Laplacian for closed Riemannian manifolds. This generalizes recent work of Kolb--Weich--Wolf [arXiv:2011.06434] on constant curvature surfaces and of Ren--Tao [arXiv:2208.13111] on locally symmetric spaces. As an application, we prove a conjecture of Baudoin--Tardif [arXiv:1604.06813] on the optimal convergence rate to the equilibrium.