Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture

• 成果类型: Article
• 署名作者: Logunov, Alexander
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2018.187.1.5
• 发表日期: 2018
• 页码: 241-262
• 关键词: equations

摘要

Let u be a harmonic function in the unit ball B(0, 1) subset of R-n, n >= 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hn-1 ({u = 0} boolean AND B ) >= c. We prove Nadirashvili's conjecture as well as its counterpart on C-infinity-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact C-infinity-smooth Riemannian manifold M (without boundary) of dimension n, there exists c > 0 such that for any Laplace eigenfunction phi lambda on M, which corresponds to the eigenvalue lambda, the following inequality holds: c root lambda <= H-n(-1)({phi lambda = 0}).