We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 over a field 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⊠Verp, where Verp is the Verlinde category (the semisimplification of Rep𝐤(ℤ/p)); a similar construction of the underlying additive functor appeared

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

We develop new techniques for studying concentration of Laplace eigenfunctions ϕλ as their frequency, λ, grows. The method consists of controlling ϕλ⁢(x) by decomposing ϕλ 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 than λ-12. We control ϕλ⁢(x) by the L2-mass of ϕλ

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

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

Elementary abelian groups are finite groups in the form of A=(ℤ/p⁢ℤ)r for a prime number p. For every integer ℓ>1 and r>1, we prove a non-trivial upper bound on the ℓ-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-torsion in class groups are bounded non-trivially for every G-extension and

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

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

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,Ke,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 satisfies this equation and touches

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 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, and Weyman on

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

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.

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B1⊂ℝ4 into the sphere 𝕊n. 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 𝕊n, which in particular yields the long-time existence

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

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, but for higher rank it

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

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 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 prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function P:ℤ≥0→ℤ, then there exists an integer N>0 such that if (X,ℱ) is a canonical or nef model of a foliation of general type with Hilbert polynomial χ⁢(X,𝒪X⁢(m⁢Kℱ))=P⁢(m) for all m∈ℤ≥0, then |m⁢Kℱ| defines a birational map for all m≥N.

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

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 764 Pages: i-iv

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)

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

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, and we recover this result and construct non-smooth solutions which are C3. We also construct solutions having prescribed behavior near the maximum. We do this by analyzing the asymptotics for rescaled mean curvature

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

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 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

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

Journal Name: Journal für die reine und angewandte Mathematik Volume: 2020 Issue: 763 Pages: i-iv

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in ℂn,n≥2, is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic

Given a stability condition on a smooth projective variety X, we construct a family of stability conditions on X×C, where C is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk

We correct a heavy misprint in our paper.

We show that any proper Lie groupoid admits a compatible real analytic structure. Our proof hinges on a Weyl unitary trick of sorts for appropriate local holomorphic groupoids.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in ℝn, whose level set {u=0} is contained in a half-space. If n≤3, we prove that the solution must be one-dimensional. In dimension n≥4, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than 12⁢(p-1) (resp. p), then the surface has negative Kodaira dimension. We also prove similar,

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in ℝ3, i.e. nonlocal minimal surfaces with a graphical structure.

Unlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for

In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show

We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([]) does not apply and we have to exploit a better understanding on the

We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes Xh. Boyarchenko’s two conjectures are on

In this paper, we prove the conjectural endoscopic character identities for tempered representations of metaplectic groups Mp⁢(2⁢n) based on the formalism of endoscopy theory by J. Adams, D. Renard and W. W. Li.

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics

We present a cohomological proof that recurrence of suitable Teichmüller geodesics implies unique ergodicity of their terminal foliations. This approach also yields concrete estimates for periodic foliations and new results for polygonal billiards.

In this paper we establish in the fast diffusion range the higher integrability of the spatial gradient of weak solutions to porous medium systems. The result comes along with an explicit reverse Hölder inequality for the gradient. The novel feature in the proof is a suitable intrinsic scaling for space-time cylinders combined with reverse Hölder inequalities and a Vitali covering argument within this

In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen

We extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence

Under the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs (X/Z,Δ) such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural

We study the Plateau problem with a lower-dimensional obstacle in ℝn. Intuitively, in ℝ3 this corresponds to a soap film (spanning a given contour) that is pushed from below by a “vertical” 2D half-space (or some smooth deformation of it). We establish almost optimal C1,12- estimates for the solutions near points on the free boundary of the contact set, in any dimension n≥2. The C1,12- estimates follow