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

TL;DR

This work studies how shrinking spectral windows $[\lambda,\lambda+(\log\lambda)^{-1}]$ reveal the sign of sectional curvature on compact manifolds by establishing sharp $L^2\to L^q$ bounds for spectral projections with $2<q\le q_c$, where $q_c=2(n+1)/(n-1)$ and $\mu(q)=n(\tfrac12-\tfrac1q)-\tfrac12$. The authors prove optimal log-quasimode estimates that improve upon universal bounds, with a $(\log\lambda)^{-1/2}$ gain in the strictly negative curvature case, a $(\log\lambda)^{-\mu(q)}$ gain for zero curvature, and no log gain for positive curvature; these bounds fail to distinguish curvature for supercritical exponents $q>q_c$. A novel microlocal decomposition into geodesic beams combined with a TT$^*$ framework and bilinear oscillatory-integral estimates, together with the Hadamard parametrix, underpins the proofs and yields sharp results for both nonpositive and negative curvature. These quantitative log-quasimode bounds enable a curvature-based classification of compact space forms for $q\in(2,q_c]$ and illuminate concentration phenomena near periodic geodesics, while also provoking new questions about optimal window sizes and possible relaxations of curvature hypotheses.

Abstract

We obtain new optimal estimates for the $L^2(M)\to L^q(M)$, $q\in (2,q_c]$, $q_c=2(n+1)/(n-1)$, 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 $n\ge2$ 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 $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any $q>q_c$.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Proposition 1
• Lemma 2
• proof
• Proposition 3
• Proposition 4
• Lemma 5
• proof : Proof of Proposition \ref{['smallprop']}
• proof : Proof of subcritical estimates in Theorem \ref{['thm1.1']}