Volume 2021, Issue 776

The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.

In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito’s logarithmic ramification filtration. This provides a positive answer to the main conjecture in [24] for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.

We describe explicitly the Howe correspondence for the symplectic-orthogonal and unitary dual pairs over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.

Given a linear group G over a field k , we define a notion of index and residue of an element g∈G⁢(k⁢((t))){g\in G(k(\kern-1.707165pt(t)\kern-1.707165pt))}. The index r⁢(g){r(g)} is a rational number and the residue a group homomorphism res(g):𝔾a⁢ or ⁢𝔾m→G{\mathop{\rm res}\nolimits(g):\mathbb{G}_{a}\text{ or }\mathbb{G}_{m}\to G}. This provides an alternative proof of Gabber’s theorem stating that G has no subgroups isomorphic to 𝔾a{\mathbb{G}_{a}} or 𝔾m{\mathbb{G}_{m}} iff G⁢(k⁢[[t]])=G⁢(k⁢((t))){G(k[\kern-1.13811pt[t]\kern-1.13811pt])=G(k(\kern-1.707165pt(t)\kern-1.707165% pt))}. In the case of a reductive group, we offer an explicit connection with the theory of affine Grassmannians.

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension ≥2{\geq 2}, can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disc. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa–Hoffman–Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover by using the min-max method.

Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L -functions. In this paper we prove that central values of additive twists of the L -function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k≥4{k\geq 4}) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L -functions consisting of central values of the form L⁢(f⊗χ,1/2){L(f\otimes\chi,1/2)} with χ a Dirichlet character.

Let 𝕍{{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S . In this paper, we show that the union of the non-factor special subvarieties for (S,𝕍){(S,{\mathbb{V}})}, which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S . This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.