Math Conjectures

Yau nodal-set conjecture

OPENMajorConjectureProposed 1982 · Canonical special case

Canonical statement

For every closed smooth Riemannian nn-manifold (M,g)(M,g) there are constants cg,Cg>0c_g,C_g>0 such that every nonzero real eigenfunction φ\varphi satisfying −Δgφ=λφ-\Delta_g\varphi=\lambda\varphi, λ>0\lambda>0, obeys
cgλ ≤ Hn−1 ⁣({x∈M:φ(x)=0}) ≤ Cgλ. c_g\sqrt{\lambda}\ \le\ \mathcal H^{n-1}\!\left(\{x\in M:\varphi(x)=0\}\right) \ \le\ C_g\sqrt{\lambda}.
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For every closed smooth Riemannian \(n\)-manifold \((M,g)\) there are constants \(c_g,C_g>0\) such that every nonzero real eigenfunction \(\varphi\) satisfying \(-\Delta_g\varphi=\lambda\varphi\), \(\lambda>0\), obeys \[ c_g\sqrt{\lambda}\ \le\ \mathcal H^{n-1}\!\left(\{x\in M:\varphi(x)=0\}\right) \ \le\ C_g\sqrt{\lambda}. \]
GEO-TOP-013:[Yau1982ProblemNodalSets][DonnellyFefferman1988NodalSets][Logunov2018NodalHypersurfaces][Logunov2018NodalLowerBound]
Geometry and topologyPartial differential equationsAnalysis and functional analysis

Notes

In his 1982 problem section Yau conjectured that the nodal set {φ=0}\{\varphi=0\} of a Laplace eigenfunction on a closed smooth Riemannian nn-manifold has (n−1)(n-1)-dimensional Hausdorff measure comparable to λ\sqrt\lambda, where λ\lambda is the eigenvalue: there should be bounds cgλ≤Hn−1({φ=0})≤Cgλc_g\sqrt\lambda\le\mathcal H^{n-1}(\{\varphi=0\})\le C_g\sqrt\lambda with constants depending only on the manifold [Yau1982ProblemNodalSets]. The heuristic is that an eigenfunction with eigenvalue λ\lambda oscillates on wavelength scale 1/λ1/\sqrt\lambda, so its zero set should resemble a hypersurface of total measure of order λ\sqrt\lambda.

Donnelly and Fefferman proved both bounds in 1988 when the metric is real-analytic [DonnellyFefferman1988NodalSets]. For merely smooth metrics the picture was transformed by Logunov, who proved the sharp lower bound cgλc_g\sqrt\lambda in all dimensions, settling Nadirashvili's conjecture along the way [Logunov2018NodalLowerBound], and established an upper bound of polynomial form CgλαC_g\lambda^{\alpha} for some exponent α\alpha [Logunov2018NodalHypersurfaces].

What remains is to replace the polynomial upper estimate by the sharp CgλC_g\sqrt\lambda for smooth metrics; this half of the conjecture is still open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.