Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

By Nicolas Burq, Patrick Gérard, Nikolay Tzvetkov. 2005

185 citations recorded by the source corpus.

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/7e8edb3558e538eaf5acf8db37bba2af4942dc6e

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Improvement of eigenfunction estimates on manifolds of nonpositive curvature (2012, methodology): “In the high p range Chen [5] obtains logarithmic improvements to the restriction theorems of Burq, Gérard and Tvetkov [4].”
Extensions of the Stein-Tomas theorem (2010, methodology): “Given the preparations in that work the proof of Theorem 1.4 is very similar to the proof of Theorem 1.2.”
EIGENFUNCTIONS AND NODAL SETS (2012, methodology): “enfunctions to hypersurfaces or higher codimension submanifolds. For expository purposes we only consider geodesics on surfaces here. Following earlier work of A. Reznikov, Burq, Gerard and Tzvetkov [BGT] proved Theorem 5.2. [BGT] Suppose that (M;g) is a compact surface, then there exists 0();C>”
Lp norm estimates of eigenfunctions restricted to submanifolds (2009, background): “(4) 38 (2005), 255–301 [4] Burq N., Gérard P., Tzvetkov N.: Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds.”
Quantum Ergodic Restriction Theorems: Manifolds Without Boundary (2011, background): “,γHϕjiL2(H)|≤CkγHχ0ϕjkL2(H)kγHϕjkL2(H) = O(λ −∞ j ). QUANTUM ERGODIC RESTRICTION THEOREMS, II: MANIFOLDS WITHOUT BOUNDARY 43 Here, one can use the universal restriction bound kγϕjkL2(H) = O(λ 1/4 j ) [BGT] to bound the kγHϕjkL2(H)-term on the right side of the Schwarz inequality, but any crude polyn”
Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions (2015, background): “Moreover, it represents a logarithmic improvement over the two-dimensional restriction estimates of Burq, Gérard and Tzvetkov [6].”
The subconvexity problem for GL2 (2009, unknown): “Geodesic restriction problems: the results of [11].”