We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n \geq 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultaneously the surface area, the volume and the boundary momentum of convex sets. As a by-product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue.

A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in dual pairs. Looijenga proved in 1981 that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. In 1983, the second author and Miranda gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K‑trivial semistable model for the central fiber of such a smoothing. We study these “Type III degenerations” of rational surfaces with an anticanonical divisor—their deformations, birational geometry, and monodromy. Looijenga’s original paper also gave a description of the rational double point configurations to which a cusp singularity deforms, but only in the case where the resolution of the dual cusp has cycle length $5$ or less. We generalize this classification to an arbitrary cusp singularity, giving an explicit construction of a semistable simultaneous resolution of such an adjacency. The main tools of the proof are (1) formulas for the monodromy of a Type III degeneration, (2) a construction via surgeries on integral-affine surfaces of a degeneration with prescribed monodromy, (3) surjectivity of the period map for Type III central fibers, and (4) a theorem of Shepherd–Barron producing a simultaneous contraction to the adjacency.

In [HLY1], Hosono, Lian, and Yau gave a conjectural characterization of the set of solutions to certain Gelfand–Kapranov–Zelevinsky hypergeometric equations which are realized as periods of Calabi–Yau hypersurfaces in a Gorenstein Fano toric variety $X$. We prove this conjecture in the case where $X$ is a complex projective space.

We solve the Fu–Yau equation for arbitrary dimension and arbitrary slope $\alpha^\prime$. Actually we obtain at the same time a solution of the open case $\alpha^\prime \gt 0$, an improved solution of the known case $\alpha^\prime \lt 0$, and solutions for a family of Hessian equations which includes the Fu–Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.