Abstract
Let (M, g) be a compact Riemannian surface with nonpositive sectional curvature and let γ be a closed geodesic in M. And let eλ be an L2-normalized eigenfunction of the Laplace–Beltrami operator Δ g with - Δ geλ= λ2eλ. Sogge et al. (Camb J Math 5(1):123–151, 2017) showed using the Gauss–Bonnet Theorem that ∫γeλds=O((logλ)-1/2), an improvement over the general O(1) bound. We show this integral enjoys the same decay for a wide variety of curves, where M has nonpositive sectional curvature. These are the curves γ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to γ.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 3204-3232 |
| Number of pages | 29 |
| Journal | Journal of Geometric Analysis |
| Volume | 30 |
| Issue number | 3 |
| DOIs | |
| State | Published - Jul 1 2020 |
Funding
This manuscript is an adapted version of the author\u2019s Ph.D. thesis. As such, the author would like to extend his deepest thanks to his advisor, Christopher Sogge, for providing the initial problem and for his continued support. The author would also like to extend thanks to Yakun Xi and Cheng Zhang, who proofread an early draft of this result and, along with Sogge, produced the paper upon which this result is based. Finally, thanks to the anonymous referee for the invaluable feedback. This work is supported in part by NSF Grant DMS-1069175.
Keywords
- Eigenfunction restrictions
- Nonpositive curvature
- Period integrals
ASJC Scopus subject areas
- Geometry and Topology