Curvature and sharp growth rates of log-quasimodes on compact manifolds

• 成果类型: Article
• 署名作者: Huang, Xiaoqi; Sogge, Christopher D.
• 署名单位: Louisiana State University System; Louisiana State University; Johns Hopkins University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01315-2
• 发表日期: 2025
• 页码: 947-1008
• 关键词: kakeya-nikodym bounds; oscillatory integrals; l-p; eigenfunctions; RESTRICTION; resolvent; Operators; THEOREM; PROOF

摘要

We obtain new optimal estimates for the L-2(M)-> L-q(M), q is an element of(2,q(c)], q(c)=2(n+1)/(n-1), operator norms of spectral projection operators associated with spectral windows [lambda,lambda+delta(lambda)], with delta(lambda)=O((log lambda)(-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 L-q-norms of quasimodes for each Lebesgue exponent q is an element of(2,q(c)], even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any q>q(c).