Volume 2016, Issue 710

Let 𝔪∈ℕ${\mathfrak {m}\in \mathbb {N}}$, 𝔪≥2${\mathfrak {m}\ge 2}$, and let {pj}j=1𝔪${\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ be a finite subset of 𝕊 2 such that 0→∈ℝ3${\vec{0}\in \mathbb {R}^3}$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies 𝒦 in ℝ 3 whose boundary surface consists of an open surface S with constant Gauss curvature (respectively, constant mean curvature) and 𝔪 planar compact discs D¯1,...,D¯𝔪${\overline{D}_1,\ldots ,\overline{D}_\mathfrak {m}}$, such that the Gauss map of S is a homeomorphism onto 𝕊2-{pj}j=1𝔪${\mathbb {S}^2-\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ and Dj⊥pj${D_j\bot p_j}$, for all j.${j.}$ We derive applications to existence of harmonic diffeomorphisms between domains of 𝕊 2 , existence of capillary surfaces in ℝ 3 , and a Hessian equation of Monge–Ampère type.

In this paper we give a description of the coefficients of the asymptotic expansion of the logarithmic derivative of a family of hypergeometric series. This family plays an important role in the computation of the reduced genus one Gromov–Witten invariants of projective hypersurfaces and the confirmation of Bershadsky, Cecotti, Ooguri, Vafa (BCOV) conjecture for genus one Gromov–Witten invariants of a generic quintic threefold by Zinger.

We establish a C 1,α -compactness theorem for metrics with bounded self-dual Weyl tensor and scalar curvature. The main ingredient is the C 1,α -harmonic radius estimate, where a blow up analysis as in [Invent. Math. 102 (1990), 429–445] is used. Our result is motivated by, and may be applied to, the Calabi flow on complex surfaces.

We show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kähler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterion expressed via the Hilbert polynomial to the Kähler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kähler manifold.

We study different notions of quasiconvexity for a subgroup H of a relatively hyperbolic group G . Our first result implies that relative geometric quasiconvexity is equivalent to dynamical quasiconvexity as it was conjectured by D. Osin. In the second part of the paper we prove that a subgroup H of a finitely generated relatively hyperbolic group G acts cocompactly outside its limit set if and only if it is (absolutely) quasiconvex and every infinite intersection of H with a parabolic subgroup of G has finite index in the parabolic subgroup. We also discuss relations between other properties of subgroups close to quasiconvexity.

Given a discrete group G${\mathrm {G}}$ and an orthogonal action γ:G→O(n)${\gamma : \mathrm {G}\rightarrow {\rm O}(n)}$ we study L p -convergence of Fourier integrals which are frequency supported on the semidirect product ℝn⋊γG${\mathbb {R}^n \rtimes _\gamma \mathrm {G}}$. Given a unit u∈ℝn${u \in \mathbb {R}^n}$ and 1<p≠2<∞${1 &lt; p \ne 2 &lt; \infty }$, our main result shows that the twisted (directional) Hilbert transform Hu⋊γ id G${H_u \rtimes _\gamma {\rm id}_\mathrm {G}}$ is L p -bounded iff the orbit 𝒪γ(u)${\mathcal {O}_\gamma (u)}$ is finite. This is in sharp contrast with twisted Riesz transforms Ru⋊γ id G${R_u \rtimes _\gamma {\rm id}_\mathrm {G}}$, which are always bounded. Our result characterizes Fourier summability in L p for this class of groups. We also extend de Leeuw's compactification theorem to this setting and obtain stronger estimates for functions with “lacunary” frequency support.

Soient k un corps de nombres et X une k -variété affine lisse intègre fibrée au-dessus de la droite affine 𝔸k1$\mathbb {A}^1_{k}$. Supposons que toutes les fibres sont géométriquement intègres, et que la fibre générique est un espace homogène sous un groupe semisimple simplement connexe presque simple G/k(t)$G/k(t)$, les stabilisateurs géométriques étant réductifs connexes. Soit v une place de k telle que la fibration X→𝔸k1$X \rightarrow \mathbb {A}^1_{k}$ admette une section rationnelle sur le complété k v . Supposons en outre que pour presque tout point x∈𝔸1(kv)$x \in \mathbb {A}^1(k_{v})$ le k v -groupe G x est isotrope. Supposons enfin le groupe de Brauer de X réduit à celui de k . Alors l'approximation forte vaut pour X en dehors de la place v . Let k be a number field and X a smooth integral affine variety equipped with a surjective morphism f:X→𝔸k1$f: X \rightarrow \mathbb {A}^1_{k}$ to the affine line. Assume that all fibres of f are split, for instance that they are geometrically integral. Assume that the generic fibre of f is a homogeneous space of a simply connected, almost simple, semisimple group G/k(t)$G/k(t)$, and that the geometric stabilizers are connected reductive groups. Let v be a place of k such that the fibration f acquires a rational section over the completion k v at v . Assume moreover that at almost all points in x∈𝔸1(kv)$x \in \mathbb {A}^1(k_{v})$ the specialized group G x is isotropic over k v . If the Brauer group of X is reduced to the Brauer group of k , then strong approximation holds for X away from the place v .

The residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. In this paper, we construct groups with arbitrarily large residual finiteness growth. We also demonstrate a new relationship between residual finiteness growth and some decision problems in groups, which we apply to our new groups.

We study the regular action of an analytic pseudo-group of transformations on the space of germs of various analytic objects of local analysis and local differential geometry. We fix a homogeneous object F 0 and we are interested in an analytic normal form for the whole affine space {F0+h.o.t.}${\lbrace F_0 + \text{h.o.t.}\rbrace }$. We prove that if the cohomological operator defined by F 0 has the big denominators property and if a formal normal form is well chosen, then this formal normal form holds in analytic category. We also define big denominators in systems of nonlinear PDEs and prove a theorem on local analytic solvability of systems of nonlinear PDEs with big denominators. Moreover, we prove that if the denominators grow “relatively fast”, but not fast enough to satisfy the big denominators property, then we have a normal form, respectively local solvability of PDEs, in a formal Gevrey category. We illustrate our theorems by explanation of known results and by new results in the problems of local classification of singularities of vector fields, non-isolated singularities of functions, tuples of germs of vector fields, local Riemannian metrics and conformal structures.