Riemannian manifolds with maximal eigenfunction growth

By Christopher D. Sogge, Steve Zelditch. 2001

184 citations recorded by the source corpus.

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. Our main result is that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We show that if $(M, g)$ is real analytic, this puts topological restrictions on $M$, e.g. only $M = S^2$ (topologically) in dimension 2 can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any $M$ fail to have maximal eigenfunction growth. In addition, we construct an example of $(M, g)$ for which ${\mathcal L}_x$ has positive measure for an open set of $x$ but which does not have maximal eigenfunction growth, thus disproving a naive converse to the main result.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/18ec48c9808ed7e7733a57520c5754a94af29629

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Citations in the corpus

Lp norm estimates of eigenfunctions restricted to submanifolds (2009, background): “Another way is by studying the growth of the Lp norms of the eigenfunctions on the manifold (see the works by Sogge [15, 16], Sogge-Zelditch [17], Burq, Gérard, and Tzvetkov [2, 3, 1]).”
On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature (2011, unknown): “4) can be improved for p > 6 (see [21], [22]) to bounds of the form ‖eλ‖Lp(M)/‖eλ‖L2(M) = o(λ 1 2 ( 1 2 − 1 p ) for fixed p > 6.”
Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds (2005, background): “Another way is the study of the potential growth of ‖φλ‖Lp(M), see the works by Sogge [18, 19], Sogge-Zelditch [20], the authors [3, 2, 4].”
Focal points and sup-norms of eigenfunctions (2013, methodology): “As in [25], it uses an idea coming from Ivrii’s [12] proof of his generalization of the Duistermaat-Guillemin theorem [7].”
Nonharmonic Analysis of Boundary Value Problems (2015, background): “Finding estimates for the norms ‖uξ‖L∞ in terms of the corresponding eigenvalues of L is a challenging problem even for self-adjoint operators L, see, for example, Sogge and Zelditch [68] and references therein.”
Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions (2009, methodology): “chy data to the solution of the wave equation restricted to R is a Fourier integral operator with a one-sided fold. Using this fact and the standard method of long-time averages (see e.g. [10], [16], [30], [29]), we shall be able to prove Proposition 3.1. To set up our proof, let us choose Fermi nor”
Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds (2004, background): “However the sharpness of these bounds is a much more difficult problem (see Sogge-Zelditch [30] for recent results in this direction), except in specific geometries.”
Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions (2015, background): “Over the years, there have been many works on improving L-norms for relatively large exponents p > 2(n+1) n−1 , including [15], [23], [24], [28] and [29].”
Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law (2014, methodology): “As in [35] we choose a real-valued function ψ̃ε ∈ C∞ c (Rn) with supp(ψ̃ε) ⊂ Nε and equal to 1 in a neighborhood of κ−1 x0 (x0).”
Fourier multipliers, symbols, and nuclearity on compact manifolds (2014, background): “asis as opposed to estimates for a favourable one that may exist due to additional assumptions or structures. However, the latter one seems to be the question much less studied in the literature, see [SZ02] or [TZ02] for some partial discussions. We now give an example of the above remark in the cas”
Improvement of eigenfunction estimates on manifolds of nonpositive curvature (2012, background): “Sogge and Zelditch [15] and Sogge, Zelditch and Toth [14] investigated the conditions on M required for maximal L∞ growth.”