Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
Let \((M,g)\) be a compact smooth Riemannian manifold of the dimension \(d\) without boundary and \(\Delta\) the associated with the metric \(g\) Laplace-Beltrami operator. Let \(\{ \varphi_\lambda \}\), \(\lambda \geq 0\) be the eigenfunctions of \(\Delta\), i.e. \(-\Delta\varphi_\lambda=\lambda^2\varphi_\lambda\). The authors study the possible growth of the \(L^p\)-norm, \(2 \leq p \leq +\infty\) of the restrictions of \(\varphi_\lambda\) to submanifolds \(\Sigma\) of \(M\) in the simplest case when \(\Sigma\) is a smooth curve \(\gamma :[a,b] \to M\) parametrized by arc length. When the curve is a geodetic it is shown that on the sphere the obtained estimates are sharp, optimal for the sphere.
restrictions of eigenfunctions to a curve
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