Eigenfunctions with few critical points.
The authors consider critical points of eigenfunctions of the Laplacian on the the two-dimensional torus with a metric of the form \[ g = q(x)(dx^2 + dy^2). \] In particular, they show that there exists a function \(q\) and a sequence of eigenvalues of the Laplacian with respect to \(g\) such that the corresponding eigenvalues converge to infinity but the number of critical points of the eigenfunctions remains bounded. This result is a negative answer of the question raised by \textit{Shing-Tung Yau} [Ann. Math. Stud. 102, 669--706 (1982; Zbl 0479.53001)]. Yau suggested that the number of critical points increases with the growth of the eigenvalue.
scientific article; zbMATH DE number 1782620