Band 2022, Heft 791

We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.

The purpose of this article is to newly define the p -adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of p -adic L -functions interpolating nonpositive values of Hecke L -functions of the totally real field in terms of special values of these p -adic polylogarithms.

We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is ≤ 2 {\leq 2} , as well as in various instances where it is 3. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel’s hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer–Manin obstruction in families.

In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley-type restriction theorem, Integrable systems, geometry, and topology, American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng, Ancient solutions to mean curvature flow for isoparametric submanifolds, Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres.

In this note we compare two ways of measuring the n -dimensional “flatness” of a set S ⊂ ℝ d {S\subset\mathbb{R}^{d}} , where n ∈ ℕ {n\in\mathbb{N}} and d > n {d>n} . The first is to consider the classical Reifenberg-flat numbers α ⁢ ( x , r ) {\alpha(x,r)} ( x ∈ S {x\in S} , r > 0 {r>0} ), which measure the minimal scaling-invariant Hausdorff distances in B r ⁢ ( x ) {B_{r}(x)} between S and n -dimensional affine subspaces of ℝ d {\mathbb{R}^{d}} . The second is an “intrinsic” approach in which we view the same set S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers 𝖺 ⁢ ( x , r ) {{\mathsf{a}}(x,r)} that are the scaling-invariant Gromov–Hausdorff distances between balls centered at x of radius r in S and the n -dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers 𝖺 ⁢ ( x , r ) {{\mathsf{a}}(x,r)} behaves as the square of the numbers α ⁢ ( x , r ) {\alpha(x,r)} . Moreover, we show how this result finds application in extending the Cheeger–Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones’ numbers β (i.e. the one-sided version of the numbers α).

We construct a pointwise Boutet de Monvel–Sjöstrand parametrix for the Szegő kernel of a weakly pseudoconvex three-dimensional CR manifold of finite type assuming the range of its tangential CR operator to be closed; thereby extending the earlier analysis of Christ. This particularly extends Fefferman’s boundary asymptotics of the Bergman kernel to weakly pseudoconvex domains in ℂ 2 {\mathbb{C}^{2}} , in agreement with D’Angelo’s example. Finally, our results generalize a three-dimensional CR embedding theorem of Lempert.

In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.

In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k {CY\times\mathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2\leq k\leq n} ) in a non-collapsing metric cone C ⁢ X × ℝ n - k {CX\times\mathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY . Namely, Y has only one connected component in X .

In this paper, we answer in the affirmative the long-standing conjecture that the first cohomology group of the Murray–von Neumann algebra S ⁢ ( ℳ ) {S(\mathcal{M})} of all operators affiliated with a type II 1 {\mathrm{II}_{1}} von Neumann algebra ℳ {\mathcal{M}} is 0. That is, we show that all derivations of S ⁢ ( ℳ ) {S(\mathcal{M})} are inner.