Cubic moments of eigenfunctions on surfaces

Source abstract

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $λ$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(λ^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\logλ)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.