Volume 2020, Issue 763

For a set of primes 𝒫{\mathcal{P}}, let Ψ⁢(x;𝒫){\Psi(x;\mathcal{P})} be the number of positive integers n≤x{n\leq x} all of whose prime factors lie in 𝒫{\mathcal{P}}. In this paper we classify the sets of primes 𝒫{\mathcal{P}} such that Ψ⁢(x;𝒫){\Psi(x;\mathcal{P})} is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki [A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 2015, 10, 1935–1969] and their main conjecture is proved in this paper. In particular, our main theorem implies that, if not too many large primes are sieved out in the sense that ∑p∈𝒫x1/v<p≤x1/u1p≥1+εu,\sum_{\begin{subarray}{c}p\in\mathcal{P}\\ x^{1/v}<p\leq x^{1/u}\end{subarray}}\frac{1}{p}\geq\frac{1+\varepsilon}{u}, for some ε>0{\varepsilon>0} and v≥u≥1{v\geq u\geq 1}, then Ψ⁢(x;𝒫)≫ε,vx⁢∏p≤xp∉𝒫(1-1p).\Psi(x;\mathcal{P})\gg_{\varepsilon,v}x\prod_{\begin{subarray}{c}p\leq x\\ p\notin\mathcal{P}\end{subarray}}\bigg{(}1-\frac{1}{p}\bigg{)}.

We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin–Abreu–Donaldson geometry of toric varieties. Namely we associate convex functions with Hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge–Ampère volume form and scalar curvature. We provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar to one obtained by Li, Zhou and Zhu on group compactifications. This finally translates to a sufficient criterion of existence of constant scalar curvature Kähler metrics thanks to the recent work of Chen and Cheng. It yields infinitely many new examples of explicit Kähler classes admitting cscK metrics.

We give asymptotics for the level set equation for mean curvature flow on a convex domain near the point where it attains a maximum. It is known that solutions are not necessarily C3{C^{3}}, and we recover this result and construct non-smooth solutions which are C3{C^{3}}. We also construct solutions having prescribed behavior near the maximum. We do this by analyzing the asymptotics for rescaled mean curvature flow converging to a stationary sphere.

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

Let p be a rational prime and N a positive integer which is prime to p . Let 𝒲{\mathcal{W}} be the p -adic weight space for GL2,ℚ{{\mathrm{GL}}_{2,\mathbb{Q}}}. Let 𝒞N{\mathcal{C}_{N}} be the p -adic Coleman–Mazur eigencurve of tame level N . In this paper, we prove that any irreducible component of 𝒞N{\mathcal{C}_{N}} which is of finite degree over 𝒲{\mathcal{W}} is in fact finite over 𝒲{\mathcal{W}}. Combined with an argument of Chenevier and a conjecture of Coleman–Mazur–Buzzard–Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.

We apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of ℝ3{\mathbb{R}^{3}}. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in ℝ3{\mathbb{R}^{3}}, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in ℝ3{\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.