Document Type
Article
Publication Date
3-1-2025
Abstract
We obtain new optimal estimates for the L2(M)→Lq(M), q∈(2,qc], qc=2(n+1)/(n−1), operator norms of spectral projection operators associated with spectral windows [λ,λ+δ(λ)], with δ(λ)=O((logλ)−1) on compact Riemannian manifolds (M,g) of dimension n≥2 all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of Lq-norms of quasimodes for each Lebesgue exponent q∈(2,qc], even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any q>qc.
Publication Source (Journal or Book title)
Inventiones Mathematicae
First Page
947
Last Page
1008
Recommended Citation
Huang, X., & Sogge, C. (2025). Curvature and sharp growth rates of log-quasimodes on compact manifolds. Inventiones Mathematicae, 239 (3), 947-1008. https://doi.org/10.1007/s00222-025-01315-2