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

By Alexander Logunov. 2016

161 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c4be1dd77a69a169744e80dcd9b2ed99562ae507

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In Applied Mathematics

1. F. C. Marques
2. A. Neves
3. Antoine Song
4. M. Li
5. Marco A. M. Guaraco
6. Xin Zhou
7. S. Brendle
8. Daniel Ketover

In Computational Theory and Mathematics

1. F. C. Marques
2. A. Neves

In Mathematical Physics

1. A. Neves
2. F. C. Marques
3. M. Li
4. Antoine Song
5. S. Almaraz
6. M. Mars

Citations in the corpus

• Existence of infinitely many minimal hypersurfaces in positive Ricci curvature (2013, result): “This conjecture was proven by Donnelly and Fefferman [8] when the metric is analytic and the lower bound has been recently proved for smooth metrics by Logunov [26].”