Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture. (English) Zbl 1384.58021

Logunov, Alexander

Ann. Math. (2) 187, No. 1, 241-262 (2018).

Let \(\mathcal{M}=(M,g)\) be a compact closed smooth Riemannian manifold of dimension \(n\). Let \(\Delta\) be the scalar Laplacian and let \(\phi_\lambda\) be a non-trivial eigenfunction of the Laplacian. The author shows that there exists \(c(\mathcal{M})>0\) so that \(c\lambda^{\frac12}\leq H^{n-1}\{x:\phi_\lambda(x)=0\}\) where \(H^{n-1}\) denotes \((n-1)\) dimensional Hausdorf measure. An analogous upper bound is established in a previous paper by the author. The author also establishes a similar estimate for harmonic functions in the unit ball of \(\mathbb{R}^n\) which establishes a conjecture by N. Nadirashvili [A. Jaffe et al., in: Current developments in mathematics, 1997. Papers from the conference held in Cambridge, MA, USA, 1997. Boston, MA: International Press. 189–266 (1999; Zbl 1031.00501)].
The introduction states the main results and provides historical contextualization. The second section treats almost monotonicity of the frequency. The third section gives a lemma concerning monotonic functions. The fourth section discusses the behaviour near the maximum. The fifth section deals with the number of cubes with big doubling index. The sixth section contains a geometrical construction which is useful for providing lower estimates for the nodal sets. The seventh section estimates the volume of the nodal set and establishes the desired estimate on the ball. The eighth section provides a lower bound in Yau’s conjecture and deals with a general Riemannian manifold.

Reviewer: Peter B. Gilkey (Eugene)

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P05	General topics in linear spectral theory for PDEs
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

harmonic functions; Laplace eigenfunctions; nodal sets; Yau’s conjecture; doubling index; frequency

Citations:

Zbl 1031.00501

References:

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