TY - JOUR

T1 - Period integrals in nonpositively curved manifolds

AU - Wyman, Emmett L.

N1 - Funding Information:
This article was supported in part by NSF grant DMS-1665373.
Publisher Copyright:
© 2020 International Press of Boston, Inc.. All rights reserved.

PY - 2021

Y1 - 2021

N2 - Let M be a compact Riemannian manifold without boundary. We investigate the integrals of L2-normalized Laplace eigenfunctions over closed submanifolds. General bounds for these quantities were obtained by Zelditch [23], and are sharp in the case where M is the standard sphere. However, as with sup norms of eigenfunctions, there are many interesting settings where improvements can be made to these bounds, e.g. where M is a negatively curved surface and the submanifold is a geodesic (see [6, 18]). So far, improvements in the nonpositive curvature setting have been confined to the two-dimensional case (see works of Chen and Sogge [6]; Sogge, Xi, and Zhang [18]; and the author [20, 22]). Here, we provide two theorems which extend these results into the higher dimensional setting. First, we provide an improvement of a half power of log over the standard bounds provided the submanifold has codimension 2 and M has strictly negative sectional curvature. Second, we provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. We use the usual tools, such as the Hadamard parametrix and the method of stationary phase, but critical to our argument is a computation of the Hessian of the distance function on the universal cover of M.

AB - Let M be a compact Riemannian manifold without boundary. We investigate the integrals of L2-normalized Laplace eigenfunctions over closed submanifolds. General bounds for these quantities were obtained by Zelditch [23], and are sharp in the case where M is the standard sphere. However, as with sup norms of eigenfunctions, there are many interesting settings where improvements can be made to these bounds, e.g. where M is a negatively curved surface and the submanifold is a geodesic (see [6, 18]). So far, improvements in the nonpositive curvature setting have been confined to the two-dimensional case (see works of Chen and Sogge [6]; Sogge, Xi, and Zhang [18]; and the author [20, 22]). Here, we provide two theorems which extend these results into the higher dimensional setting. First, we provide an improvement of a half power of log over the standard bounds provided the submanifold has codimension 2 and M has strictly negative sectional curvature. Second, we provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. We use the usual tools, such as the Hadamard parametrix and the method of stationary phase, but critical to our argument is a computation of the Hessian of the distance function on the universal cover of M.

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U2 - 10.4310/MRL.2020.v27.n5.a10

DO - 10.4310/MRL.2020.v27.n5.a10

M3 - Article

AN - SCOPUS:85100178500

VL - 27

SP - 1513

EP - 1563

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 5

ER -