2018/09/17 by Yaiza Canzani, Jeffrey Galkowski, Canzani, Yaiza +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1809.06296
openalex publication_date 2018/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M,g) be a smooth, compact Riemannian manifold and \ϕλ\ an L2-normalized sequence of Laplace eigenfunctions, -Δgϕλ=λ2 ϕλ. Given a smooth submanifold H ⊂ M of codimension k≥ 1, we find conditions on the pair (M,H), even when H=\x\, for which |∫HϕλdσH|=O(\fracλ(k-1)/(2)√(log λ)) or |ϕλ(x)|=O(\fracλ(n-1)/(2)√(log λ)), as λ→ ∞. These conditions require no global assumption on the manifold M and instead relate to the structure of the set of recurrent directions in the unit normal bundle to H. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if (M,g) is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any H⊂ M. We also find weaker conditions than having no conjugate points which guarantee √(log λ) improvements for the L^∞ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.