Volume 2022, Issue 784

We study Tannakian properties of the convolution product of perverse sheaves on elliptic curves. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to G2{G_{2}}. This monodromy approach generalizes a result of Katz on the existence of G2{G_{2}}-motives in the middle cohomology of deformations of Beauville surfaces.

In this paper we analyze the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.

Let X be an arithmetic variety over the ring of integers of a number field K , with smooth generic fiber XK{X_{K}}. We give a formula that relates the dimension of the first Arakelov–Chow vector space of X with the Mordell–Weil rank of the Albanese variety of XK{X_{K}} and the rank of the Néron–Severi group of XK{X_{K}}. This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ{\mathbb{R}}-divisors on X are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming a conjecture by H. Gillet and C. Soulé.

We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in PSL2⁢(ℝ){{\mathrm{PSL}_{2}(\mathbb{R})}}. We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.

We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by two invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet–Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes.

In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, completely solving an open problem due to Gromov (see Question 1.1). Then we introduce a fill-in invariant (see Definition 1.2) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds via this fill-in invariant. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov’s two conjectures formulated in [M. Gromov, Four lectures on scalar curvature, preprint 2019] (see Conjecture 1.1 and Conjecture 1.2 below).

We show that the β-numbers of intrinsic Lipschitz graphs of Heisenberg groups ℍn{\mathbb{H}_{n}} are locally Carleson integrable when n≥2{n\geq 2}. Our main bound uses a novel slicing argument to decompose intrinsic Lipschitz graphs into graphs of Lipschitz functions. A key ingredient in our proof is a Euclidean inequality that bounds the β-numbers of the original graph in terms of the β-numbers of many families of slices. This allows us to use recent work of Fässler and Orponen (2020) which asserts that Lipschitz functions satisfy a Dorronsoro inequality.