Near-boundary rigidity and strict maximum modulus for Neumann eigenfunctions on convex planar domains

We prove that the maximum modulus of every nonzero second Neumann eigenfunction on a bounded convex planar domain $\mathrm{\Omega }$ is attained only on the boundary, without boundary smoothness or eigenvalue simplicity. The key is a rigidity theorem for any positive Neumann eigenvalue $\mu$: an interior maximum-modulus point $p$ with $\sqrt{\mu }\mathrm{dist}(p,\mathrm{\partial }\mathrm{\Omega })<\pi \mathrm{/}\sqrt{3}$ forces the domain to be a rectangle and the eigenfunction to be a cosine of one coordinate. In particular, $\mu \ge 4{\mu }_2(\mathrm{\Omega })$, and the factor $4$ is sharp. This result reveals how a local distance condition at a single interior maximum-modulus point can enforce global geometric rigidity, thereby linking extremum localization to the classification of domains and eigenfunctions.