Critical sets of eigenfunctions and Yau conjecture

Details Critical sets of eigenfunctions and Yau conjecture Critical sets of eigenfunctions and Yau conjecture

2012-03-09

arxiv.org/abs/1203.2089v1

Mathematics

Differential Geometry

11 pages

Scientific paper

S.T.Yau posed a conjecture that the number of critical points of the $k$-th eigenfunction on a compact Riemannian manifold (strictly) increases with $k$. As a counterexample, Jakobson and Nadirashvili constructed a metric on 2-torus such that the eigenvalues tend to infinity whereas the number of critical points remains a constant. %But their example cannot deny the "non-decreasing" part of Yau conjecture. In this paper, The present paper finds several interesting eigenfunctions on the minimal isoparametric hypersurface $M^n$ of FKM-type in $S^{n+1}(1)$, giving a series of counterexamples to Yau conjecture. More precisely, the three eigenfunctions on $M^n$ correspond to eigenvalues $n$, $2n$ and $3n$, while their critical sets consist of 8 points, a submanifold and 8 points, respectively. On one of its focal submanifolds, a similar phenomenon occurs. However, it is possible that Yau conjecture holds true for a generic metric.

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