Cubic moments of eigenfunctions on surfaces
Abstract
We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For L2-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 L2-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.
Related
- Minimal immersions by first eigenfunctions of Riemann surfaces with large conformal automorphism groups. The Klein quartic
- Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces
- Positive Geometries from Cubic Surfaces
- Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
- Intrinsic mirror symmetry for orbifold cubic surfaces and the A1 spherical DAHA
BibTeX
@misc{wang2026cubic,
title = {{Cubic moments of eigenfunctions on surfaces}},
author = {Xing Wang and Cheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16157},
eprint = {2610.16157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16157},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cubic moments of eigenfunctions on surfaces
AU - Xing Wang
AU - Cheng Zhang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16157
UR - https://arxiv.org/abs/2610.16157
AB - 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.
ER -