Volume 2024, Issue 808

We construct minimal surfaces by gluing singly periodic Karcher–Scherk saddle towers along their wings. Such constructions were previously implemented assuming a horizontal reflection plane. We break this symmetry by prescribing phase differences between the saddle towers. It turns out that, in addition to the previously known horizontal balancing condition, the saddle towers must also be balanced under a subtle vertical interaction. This interaction vanishes in the presence of a horizontal reflection plane, hence was not perceived in previous works. Our construction will be presented in a series of papers. In this first paper of the series, we will explain the background of the project and establish the graph theoretical setup that will be useful for all papers in the series. The main task of the current paper is to glue saddle towers into triply periodic minimal surfaces (TPMSs). Our construction expands many previously known TPMSs into new 5-parameter families, therefore significantly advances our knowledge on the space of TPMSs.

We show that every classical Enriques surface containing a smooth rational curve is a Reye congruence.

We construct a class of time-symmetric initial data sets for the Einstein vacuum equations modeling elementary configurations of multiple almost isolated systems. Each such initial data set consists of a collection of several localized sources of gravitational radiation, and lies in a family of data sets which is closed under scaling out the distances between the systems by arbitrarily large amounts. This class contains data sets which are not asymptotically flat, but to which nonetheless a finite ADM mass can be ascribed. The construction proceeds by a gluing scheme using the Brill–Lindquist metric as a template. Such initial data are motivated in part by a desire to understand the dynamical interaction of distant systems in the context of general relativity. As a by-product of the construction, we produce families of complete, scalar-flat initial data with trivial topology and infinitely many minimal spheres, as well as families of initial data with infinitely many Einstein–Rosen bridges.

We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to interesting consequences, e.g., when the tuple consists of powers of a single operator. We also prove several variants of this result of independent interest. The paper substantially extends former research on matrix representations in infinite-dimensional spaces dealing mainly with prescribing the main diagonals.

Fresán, Sabbah, and Yu constructed motives M n + 1 k ⁢ ( Kl ) {\mathrm{M}_{n+1}^{k}(\mathrm{Kl})} over ℚ {\mathbb{Q}} encoding symmetric power moments of Kloosterman sums in n variables. When n = 1 {n=1} , they use the irregular Hodge filtration on the exponential mixed Hodge structure associated with M 2 k ⁢ ( Kl ) {\mathrm{M}_{2}^{k}(\mathrm{Kl})} to compute the Hodge numbers of M 2 k ⁢ ( Kl ) {\mathrm{M}_{2}^{k}(\mathrm{Kl})} , which turn out to be either 0 or 1. In this article, I explain how to compute the (irregular) Hodge numbers of M n + 1 k ⁢ ( Kl ) {\mathrm{M}_{n+1}^{k}(\mathrm{Kl})} for n = 2 {n=2} or for general values of n such that gcd ⁡ ( k , n + 1 ) = 1 {\gcd(k,n+1)=1} . I will also discuss related motives attached to Airy moments constructed by Sabbah and Yu. In particular, the computation shows that there are Hodge numbers bigger than 1 in most cases.

A conjecture of Erdős states that, for any large prime q , every reduced residue class ( mod ⁡ q ) {(\operatorname{mod}q)} can be represented as a product p 1 ⁢ p 2 {p_{1}p_{2}} of two primes p 1 , p 2 ≤ q {p_{1},p_{2}\leq q} . We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer q , every reduced residue class ( mod ⁡ q ) {(\operatorname{mod}q)} can be written as p 1 ⁢ p 2 ⁢ p 3 {p_{1}p_{2}p_{3}} with p 1 , p 2 , p 3 ≤ q {p_{1},p_{2},p_{3}\leq q} primes. We also show that, for any ε > 0 {\varepsilon>0} and any sufficiently large integer q , at least ( 2 3 - ε ) ⁢ φ ⁢ ( q ) {(\frac{2}{3}-\varepsilon)\varphi(q)} reduced residue classes ( mod ⁡ q ) {(\operatorname{mod}q)} can be represented as a product p 1 ⁢ p 2 {p_{1}p_{2}} of two primes p 1 , p 2 ≤ q {p_{1},p_{2}\leq q} . The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ℤ q × {\mathbb{Z}_{q}^{\times}} of small index and study in detail the exceptional case that there exists a quadratic character ψ ⁢ ( mod ⁡ q ) {\psi~{}(\operatorname{mod}\,q)} such that ψ ⁢ ( p ) = - 1 {\psi(p)=-1} for very many primes p ≤ q {p\leq q} .

We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is ∼ ε {\sim\varepsilon} close to satisfying the optimal density, then it is, in a suitable sense, close to the E 8 {E_{8}} and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large “frame” through which our packing locally looks like E 8 {E_{8}} or Λ 24 {\Lambda_{24}} . Our methods make explicit use of the magic functions constructed in [M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 2017, 3, 991–1015] in dimension 8 and in [H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 2017, 3, 1017–1033] in dimension 24, together with results of independent interest on the abstract stability of the lattices E 8 {E_{8}} and Λ 24 {\Lambda_{24}} .

We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Zn\mathbb{Z}_{n} grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π1\pi_{1}-injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.