Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan–Fubini-type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.

We prove global well-posedness and scattering for the defocusing cubic nonlinear wave equation on ${\mathbf{R}}^{1+3}$ with radial initial data lying in the critical Sobolev space ${\dot{H}}^{1∕2}({\mathbf{R}}^3)\times {\dot{H}}^{-1∕2}({\mathbf{R}}^3)$. This result is sharp for radial initial data.

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically ${\mathrm{G}}_2$-manifolds, that is, Riemannian 7-manifolds $(M,g)$ whose holonomy group is the compact exceptional Lie group ${\mathrm{G}}_2$. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger–Fukaya–Gromov theory of collapse with bounded curvature.

The construction starts with a complete noncompact asymptotically conical Calabi–Yau 3-fold B and a circle bundle $M\to B$ satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete ${\mathrm{G}}_2$-metrics $g_{\mathit{ϵ}}$ on M that collapses with bounded curvature as $\mathit{ϵ}\to 0$ to the original Calabi–Yau metric on the base B. The ${\mathrm{G}}_2$-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry.

We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected ${\mathrm{G}}_2$-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact ${\mathrm{G}}_2$-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact ${\mathrm{G}}_2$-metrics were known.

Let $\mathcal{M}=\mathrm{\Gamma }\setminus {\mathbb{H}}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d∕2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2({\mathrm{T}}^1(\mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(\mathcal{M})$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varjú on expanders, this implies uniform exponential mixing for congruence covers of $\mathcal{M}$ when Γ is a Zariski-dense subgroup contained in an arithmetic subgroup of ${\mathrm{SO}}^{\circ }(d,1)$.