A Dynamical Approach to the Berezin-Li-Yau Inequality

We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_\Lambda(\Omega_t)$. For convex domains we show that $R_\Lambda$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_\Lambda$ and the mean curvature $H$, established in all dimensions: in $d=2$ via a near-disk Fourier analysis, and in $d\ge 3$ via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Ces\`aro-P\'olya inequality for eigenvalue averages.