Article Frontmatter was published on July 1, 2021 in the journal Journal für die reine und angewandte Mathematik (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

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

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

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

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

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

Article Frontmatter was published on June 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 775).

We prove a priori bounds for all derivatives of non-relativistic Coulombic eigenfunctions ψ, involving negative powers of the distance to the singularities of the many-body potential. We use these to derive bounds for all derivatives of the corresponding one-electron densities ρ, involving negative powers of the distance from the nuclei. The results are both natural and optimal, as seen from the ground

Let A be a connection of a principal bundle P over a Riemannian manifold M , such that its curvature FA∈Lloc2⁢(M){F_{A}\in L_{\mathrm{loc}}^{2}(M)} satisfies the stationarity equation. It is a consequence of the stationarity that θA⁢(x,r)=ec⁢r2⁢r4-n⁢∫Br⁢(x)|FA|2⁢𝑑Vg{\theta_{A}(x,r)=e^{cr^{2}}r^{4-n}\int_{B_{r}(x)}|F_{A}|^{2}\,dV_{g}} is monotonically increasing in r , for some c depending only on

For a connected reductive group G defined over a non-archimedean local field F , we consider the Bernstein blocks in the category of smooth representations of G⁢(F){G(F)}. Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called regular Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of F is not too small. Under mild hypotheses

We classify all minimal models X of dimension n , Kodaira dimension n-1{n-1} and with vanishing Chern number c1n-2⁢c2⁢(X)=0{c_{1}^{n-2}c_{2}(X)=0}. This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ>0{\epsilon>0} such that any minimal threefold satisfies either c1⁢c2=0{c_{1}c_{2}=0} or -c1⁢c2>ϵ{-c_{1}c_{2}>\epsilon}. This

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over Рёџ{{\mathbb{Q}}}. We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition

In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p -adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the

We develop new techniques for studying concentration of Laplace eigenfunctions ϕλ{\phi_{\lambda}} as their frequency, λ, grows. The method consists of controlling ϕλ⁢(x){\phi_{\lambda}(x)} by decomposing ϕλ{\phi_{\lambda}} into a superposition of geodesic beams that run through the point x . Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower

The Hodge-FVH correspondence establishes a relationship between the special cubic Hodge integrals and an integrable hierarchy, which is called the fractional Volterra hierarchy. In this paper we prove this correspondence. As an application of this result, we prove a gap condition for certain special cubic Hodge integrals and give an algorithm for computing the coefficients that appear in the gap condition

Article Frontmatter was published on May 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 774).

In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal

We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 {{\mathscr{D}}} -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity

Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p -adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over

We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when

We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism

Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric ⊥ > 0 {\operatorname{Ric}^{\perp}>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of ( p , 0 ) {(p,0)} -forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition Ric k > 0 {\operatorname{Ric}_{k}>0}

Article Frontmatter was published on April 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 773).

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f -harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

Let M be a simply connected homogeneous three-manifold with isometry group of dimension 4, and let Σ be any compact surface of genus zero immersed in M whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation Φ ⁢ ( H , K e , K ) = 0 {\Phi(H,K_{e},K)=0} . In this paper we prove that Σ is a sphere of revolution, provided that the unique inextendible rotational surface S in M that

We give a formula of an asymptotic expansion of a function, provided in the form of a fiber integral around a critical value of a holomorphic map of toric type.

Elementary abelian groups are finite groups in the form of A = ( ℤ / p ⁢ ℤ ) r {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p . For every integer ℓ > 1 {\ell>1} and r > 1 {r>1} , we prove a non-trivial upper bound on the ℓ {\ell} -torsion in class groups of every A -extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G , the ℓ {\ell}

The determinantal variety Σ p ⁢ q {\Sigma_{pq}} is defined to be the set of all p × q {p\times q} real matrices with p ≥ q {p\geq q} whose ranks are strictly smaller than q . It is proved that Σ p ⁢ q {\Sigma_{pq}} is a minimal cone in ℝ p ⁢ q {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.

We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p , and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category

This is an investigation into the possible existence and consequences of a Birch–Swinnerton-Dyer-type formula for L -functions of elliptic curves twisted by Artin representations. We translate expected properties of L -functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich group, which do not appear to be tractable

Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L -values via Pellarin-type identities

Article Frontmatter was published on March 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 772).

We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B 1 ⊂ ℝ 4 {B_{1}\subset\mathbb{R}^{4}} into the sphere 𝕊 n {\mathbb{S}^{n}} . In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into

This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It

We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian ( M n , g ) {(M^{n},g)} manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator Δ + ( n - 2 ) ⁢ ρ {\Delta+(n-2)\rho} is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on

We determine the image and the fibers for solvable base change.

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n × n {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar,

In 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank ≤ 2 {\leq 2} , but for higher

Article Erratum to C*-simplicity of locally compact Powers groups (J. reine angew. Math. 748 (2019), 173–205) was published on March 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 772).

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Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 769 Pages: i-iv

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 768 Pages: 183-183

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 768 Pages: i-iv

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 767 Pages: i-iv

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 766 Pages: i-iv

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 765 Pages: i-iv

We study the behaviour of the Arakelov metric on a smooth curve under semistable degeneration. The final result is a complicated formula involving the local discriminants of the singularities, and the graph governing the degeneration.

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.

We construct smooth rational real algebraic varieties of every dimension ≥4 which admit infinitely many pairwise non-isomorphic real forms.

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a ℤ-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r-spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation

Let f:ℂ→X be a transcendental holomorphic curve into a complex projective manifold X. Let L be a very ample line bundle on X. Let s be a very generic holomorphic section of L and D the zero divisor given by s. We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.

In this paper we study the structure of the pointed-Gromov–Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger–Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any non-collapsing Ricci shrinker limit space is strongly convex, inspired by Colding–Naber’s original idea of parabolic

Let X be a compact Kähler manifold. Given a big cohomology class {θ}, there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮⁢(X,θ), the space of singularity types of potentials. We introduce a natural pseudo-metric d𝒮 on 𝒮⁢(X,θ) that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence