Yau's nodal-set conjecture for smooth metrics: is nodal measure comparable to sqrt(lambda)?

Dimensions three and four (Theorems 1.1, 1.2 of the principal manuscript): a smooth metric arbitrarily close to the round metric on $S^3$, and one on $S^2\times T^2$, with exact eigenfunctions whose nodal measure divided by $\sqrt\lambda$ is unbounded. Dimension five and up (companion): a metric on $S^4\times S^1$ with nodal measure $\gg\lambda^{1/2+\epsilon_0}$ for a fixed $\epsilon_0>0$, and products give every dimension above five. Surfaces (companion): $\mathcal H^1(Z_u)\le C(M,g)\sqrt\lambda$, so with Bruning's lower bound Yau's conjecture holds in dimension two. So the upper bound is true for surfaces and false for smooth metrics in every dimension $\ge3$; the lower bound (Logunov) is untouched, and real-analytic metrics are unaffected. No power excess is claimed in dimensions 3 and 4.

Dimensions three and four (Theorems 1.1, 1.2 of the principal manuscript): a smooth metric arbitrarily close to the round metric on $S^3$, and one on $S^2\times T^2$, with exact eigenfunctions whose nodal measure divided by $\sqrt\lambda$ is unbounded. Dimension five and up (companion): a metric on $S^4\times S^1$ with nodal measure $\gg\lambda^{1/2+\epsilon_0}$ for a fixed $\epsilon_0>0$, and products give every dimension above five. Surfaces (companion): $\mathcal H^1(Z_u)\le C(M,g)\sqrt\lambda$, so with Bruning's lower bound Yau's conjecture holds in dimension two. So the upper bound is true for surfaces and false for smooth metrics in every dimension $\ge3$; the lower bound (Logunov) is untouched, and real-analytic metrics are unaffected. No power excess is claimed in dimensions 3 and 4.