Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature


Author: Xuehua Chen
Journal: Trans. Amer. Math. Soc. 367 (2015), 4019-4039
MSC (2010): Primary 35F99; Secondary 35L20, 42C99
DOI: https://doi.org/10.1090/S0002-9947-2014-06158-8
Published electronically: September 18, 2014
MathSciNet review: 3324918
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (M,g)$ be an $ n$-dimensional compact boundaryless Riemannian manifold with nonpositive sectional curvature. Then our conclusion is that we can give improved estimates for the $ L^p$ norms of the restrictions of eigenfunctions of the Laplace-Beltrami operator to smooth submanifolds of dimension $ k$, for $ p>\dfrac {2n}{n-1}$ when $ k=n-1$ and $ p>2$ when $ k\leq n-2$, compared to the general results of Burq, Gérard and Tzvetkov. Earlier, Bérard gave the same improvement for the case when $ p=\infty $, for compact Riemannian manifolds without conjugate points for $ n=2$, or with nonpositive sectional curvature for $ n\geq 3$ and $ k=n-1$. In this paper, we give the improved estimates for $ n=2$, the $ L^p$ norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that the exponential map from any point in $ x\in M$ is a universal covering map from $ \mathbb{R}^2\backsimeq T_{x}M$ to $ M$, which allows us to lift the calculations up to the universal cover $ (\mathbb{R}^2,\tilde {g})$, where $ \tilde {g}$ is the pullback of $ g$ via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on $ (\mathbb{R}^2,\tilde {g})$, the stationary phase estimates, and the fact that the principal coefficient of the Hadamard parametrix is bounded, by observations of Sogge and Zelditch. The improved estimates also work for $ n\geq 3$, with $ p>\frac {4k}{n-1}$. We can then get the full result by interpolation.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35F99, 35L20, 42C99

Retrieve articles in all journals with MSC (2010): 35F99, 35L20, 42C99


Additional Information

Xuehua Chen
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: xchen@math.jhu.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06158-8
Keywords: Eigenfunction estimates, negative curvature
Received by editor(s): May 12, 2012
Received by editor(s) in revised form: November 9, 2012, and February 19, 2013
Published electronically: September 18, 2014
Additional Notes: The author would like to cordially thank her advisor, Christopher Sogge, for his generous help and unlimited patience.
Article copyright: © Copyright 2014 American Mathematical Society