## Abstract

On any compact Riemannian manifold (M, g) of dimension n, the L^{2}-normalized eigenfunctions {φλ} satisfy ∥φλ∥∞ ≤ Cλ^{(n-1)/2}, where -Δφλ = λ^{2}φλ. 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, for example, for flat tori ℝ^{n}/Γ. We say that S^{n}, but not ℝ^{n}/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem that 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 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; for example, only M = S^{2} or M = ℝP^{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 ℒ_{x} has positive measure for an open set of x but which does not have maximal eigenfunction growth; thus, it disproves a naive converse to the main result.

Original language | English (US) |
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Pages (from-to) | 387-437 |

Number of pages | 51 |

Journal | Duke Mathematical Journal |

Volume | 114 |

Issue number | 3 |

DOIs | |

State | Published - Sep 15 2002 |

## ASJC Scopus subject areas

- Mathematics(all)