Yau's nodal volume conjecture

From papers

Let (M,g)(M,g) be a compact Riemannian manifold without boundary, and let φ\varphi be an eigenfunction satisfying

Δφ=λ2φ.\Delta\varphi=-\lambda^2\varphi.

Write Lφ(M)\mathcal{L}_{\varphi}(M) for its nodal volume. Yau's conjecture. There exist constants c1(M,g),c2(M,g)>0c_1(M,g),c_2(M,g)>0 such that

c1(M,g)Lφ(M)λc2(M,g).c_1(M,g)\leq \frac{\mathcal{L}_{\varphi}(M)}{\lambda}\leq c_2(M,g).

Thus the nodal volume should be of the order of the corresponding frequency. The source provides no evidence that this conjecture has been resolved in the stated generality.

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Sources & referencesView supporting material

Primary source

Michele Stecconi and Anna Paola Todino, “Statistics on Yau's conjecture: Variance asymptotics”, arXiv:2607.09946 (2026).

Additional references

7 papers in this index state this conjecture (2011–2026). The statement above is taken from the most recent of them; the others are arXiv:2512.12305, arXiv:2508.07861, arXiv:1512.06818, arXiv:1408.7101, arXiv:1402.4323, arXiv:1112.4352.

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