Nodal sets for sums of eigenfunctions on Riemannian manifolds

Abstract

Quantitative versions of unique continuation are proved for finite sums of eigenfunctions of the Laplacian on compact Riemannian manifolds. The results include a lower bound for the order of vanishing, a growth estimate for the supremum on compact balls, and a gradient bound. For real analytic metrics, an upper bound for the Hausdorff measure of the zero set is derived.

Perceptual Metrics for Static and Dynamic Triangle Meshes

“…Recently, an explanation has been proposed for the effect of different sensitivities of the HVS to the modifications of different mesh frequency components [Tor11]. Actually, by using results from Nodal sets theory [DF88] and some simple trigonometric computations, one can obtain the intrinsic frequency of each transform basis vector (i.e. each eigenvector of the mesh Laplacian matrix) and relate this frequency to the frequency as observed by human eyes under a regular viewing condition (the observed frequency is expressed in cpd, cycles per degree).…”

Semiclassical Cauchy estimates and applications

“…When V ≡ 0, i.e. u is eigenfunctions of the Laplacian operator on M with eigenvalues Eh −2 , this is the analytic case of Yau's conjecture [18] and is proved by Donnelly-Fefferman [5]. We shall follow their argument closely.…”

“…We shall use 3.4 to prove Theorem 3.1. The tunneling estimate follows from the standard overlapping chains of balls argument introduced by Donnelly and Fefferman [5] while the doubling property is a corollary of the tunneling and the Carleman estimates on shells.…”

“…We should remark that for differential operators one can obtain estimates equivalent to (1.2) (see Proposition 2.2) by using Hörmander's approach to analytic hypoellipticity and rescaling -see [5,Lemma 7.1]. In fact, we learned about this after proving (1.2) directly using the FBI transform and the study of the Donnelly-Fefferman paper [5] led to applications to the volume of nodal sets (zero set of u(h)) in the semiclassical setting.…”

“…In fact, we learned about this after proving (1.2) directly using the FBI transform and the study of the Donnelly-Fefferman paper [5] led to applications to the volume of nodal sets (zero set of u(h)) in the semiclassical setting. An illustration of the level sets of eigenfunctions is shown in Fig.…”