Yau's nodal volume conjecture

About 15 years old · traced to

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)λ≤c2(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.

References

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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