We consider two natural classes of minimal laminations in threemanifolds. Both classes may be thought of as limits—in different senses—of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations have genus zero. This answers a question posed by Hoffman and White.

We show that every (possibly degenerate) contact form on a closed three-manifold has at least two embedded Reeb orbits. We also show that if there are only finitely many embedded Reeb orbits, then their symplectic actions are not all integer multiples of a single real number; and if there are exactly two embedded Reeb orbits, then the product of their symplectic actions is less than or equal to the contact volume of the manifold. The proofs use a relation between the contact volume and the asymptotics of the amount of symplectic action needed to represent certain classes in embedded contact homology, recently proved by the authors and V. Ramos.

We prove two types of nodal results for density $1$ subsequences of an orthonormal basis $\{ \phi_j \}$ of eigenfunctions of the Laplacian on a negatively curved compact surface $(M,g)$. The first result pertains to Riemann surfaces $(M, J, \sigma)$ with an anti-holomorphic involution $\sigma$ such that $M - \mathrm{Fix}(\sigma)$ has more than one component. In any genus $\mathfrak{g}$, there is a $(3\mathfrak{g} - 3)$-dimensional moduli space of such real Riemann surfaces. Our main result is that, for any negatively curved $\sigma$-invariant metric $\mathfrak{g}$ on $M$, the number of nodal domains of the even or odd $\Delta_{\mathfrak{g}}$-eigenfunctions tends to infinity along a density $1$ subsequence. For a generic $\sigma$-invariant negatively curved metric $\mathfrak{g}$, the multiplicity of all eigenvalues equals $1$, and all eigenfunctions are either even or odd, and therefore the result holds for almost any eigenfunction.

The analytical part of the proof shows that the number of zeros of even eigenfunctions restricted to $\mathrm{Fix}(\sigma)$, and the number of singular points of odd eigenfunctions on $\mathrm{Fix}(\sigma)$, tend to infinity. This is a quantum ergodic restriction phenomenon. Our second result generalizes this statement to any negatively curved surface $(M,g)$ and to a generic curve $C \subset M$: the number of zeros of eigenfunctions $\phi_j |_C$ tends to infinity.

The additional step to obtain a growing number of nodal domains in the $(M, J, \sigma)$ setting is topological. It generalizes an argument of Ghosh, Reznikov, and Sarnak on the modular domain to higher genus.

We establish an injective correspondence $M \to \mathcal{E}(M)$ between real-analytic nonminimal hypersurfaces $M \subset \mathbb{C}^2$, spherical at a generic point, and a class of second order complex ODEs with a meromorphic singularity. We apply this result to the proof of the bound $\dim \mathfrak{hol}(M,p) \leq 5$ for the infinitesimal automorphism algebra of an arbitrary germ $(M,p) \not \sim (S^3, p')$ of a real-analytic Levi nonflat hypersurface $M \subset \mathbb{C}^2$ (the Dimension Conjecture). This bound gives the proof of the dimension gap $\dim \mathfrak{hol}(M,p) = \{8, 5, 4, 3, 2, 1, 0 \}$ for the dimension of the automorphism algebra of a real-analytic Levi nonflat hypersurface. As another application we obtain a new regularity condition for CR-mappings of nonminimal hypersurfaces, that we call Fuchsian type, and prove its optimality for the extension of CR-mappings to nonminimal points.

We also obtain an existence theorem for solutions of a class of singular complex ODEs.

We prove that the Gromov–Hausdorff compactification of the moduli space of Kähler–Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian’s theorem on the existence of Kähler–Einstein metrics on smooth Del Pezzo surfaces and classifies all the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.