Volumes for eigensections
Let \(M\) be a closed \(n\)-dimensional manifold and \(\{ f_i\}\) an \(L^2\)-orthonormal set of eigensections of a vector bundle over \(M\) for the Laplacian with eigenvalues \(\lambda_i\) (where \(0= \lambda_0 \leq \lambda_1 \leq \dots\)). Given \(a =(a_1, \dots , a_i)\in S^{i-1}\) one defines a function \(F^a_i(\varepsilon)\) as the volume of the set of points \(x\) in \(M\) such that \(| \sum_{j=1}^{i} a_j f_j(x) | <\varepsilon\) and a function \(F^A_i(\varepsilon)\) as the average of the \(F^a_i(\varepsilon)\) over \(S^{i-1}\). Roughly speaking, \(F^A_i(\varepsilon)\) represents the volume of the sets where the eigenfunctions are small. S. T. Yau posed the problem of investigating the behaviour of these functions \(F^a_i(\varepsilon)\) and \(F^A_i(\varepsilon)\). In the present paper, the authors show that \(F_i(\varepsilon):=F^{e_i}_i(\varepsilon)\) is positive and bounded uniformly from below. Moreover they show that the average \(F^A_i(\varepsilon)\) for eigenforms is positive and they give a sharp lower bound.
scientific article; zbMATH DE number 2037036