Power-law violations of Yau's nodal upper bound
We construct a smooth Riemannian metric on $S^4\times S^1$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than $\lambda^{1/2+\epsilon_0}$ for one fixed $\epsilon_0\gt 0$. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
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