Upper bound direction of Yau’s nodal-set conjecture

Prove the upper bound in Yau’s conjecture: establish that for every Laplace eigenfunction satisfying $-Delta_gvarphi_\lambda=\lambda\varphi_\lambda$ on a closed smooth $n$-dimensional Riemannian manifold $(M,g)$, its nodal hypersurface satisfies $H^{n-1}(\{\varphi_\lambda=0\})\leq C_g\sqrt{\lambda}$, with $C_g$ depending only on the Riemannian metric and not on the eigenvalue.

Background

Yau’s conjecture predicts two-sided nodal-volume estimates for Laplace eigenfunctions on closed smooth Riemannian manifolds: both the lower and upper bounds should be proportional to the square root of the eigenvalue. The paper notes that Donnelly and Fefferman proved the full conjecture for real-analytic metrics, while the sharp lower bound has subsequently been established in the smooth category in all dimensions. The upper bound remains unresolved in the smooth setting, although the paper cites several works that provide partial progress.