We prove an extension theorem for effective purely log-terminal pairs (X, S + B) of non-negative Kodaira dimension ${\kappa (K_X+S+B)\ge 0}$ . The main new ingredient is a refinement of the Ohsawa–Takegoshi L² extension theorem involving singular Hermitian metrics.

We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $${(-\mathrm{\Delta })}^{s}Q+Q-Q^{\mathit{\alpha }+1}=0\mathrm{i}n\mathbb{R},$$where 0 < s < 1 and 0 < α < 4s/(1−2s) for and 0 < α < ∞ for $s\ge \frac{1}{2}$. Here (−Δ)^s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=\frac{1}{2}$ and α = 1 in [5] for the Benjamin–Ono equation.

As a technical key result in this paper, we show that the associated linearized operator L₊ = (−Δ)^s+1−(α+1)Q^α is non-degenerate; i.e., its kernel satisfies ker L₊ = span{Q′}. This result about L₊ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n^−1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

In our previous paper “The primes contain arbitrarily long polynomial progressions” [Acta Math., 201 (2008), 213–305] there were some minor errors in our definition of the polynomial forms and polynomial correlation conditions, which is unreasonably strong as written due to the overly loose bound on the degree of the polynomials involved. In this erratum we repair this issue by capping the degree of the polynomials more strongly, and adjusting some other relevant components of the argument accordingly.