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  • Ancient solutions to the Ricci flow in dimension $3$
    Acta Math. (IF 2.458) Pub Date : 2020-09-01
    Simon Brendle

    It follows from work of Perelman that any finite-time singularity of the Ricci flow on a compact $3$-manifold is modeled on an ancient $\varkappa$-solution. We prove that every non-compact ancient $\varkappa$-solution in dimension $3$ is isometric to a family of shrinking cylinders (or a quotient thereof), or to the Bryant soliton. This confirms a conjecture of Perelman.

  • Rigid connections and $F$-isocrystals
    Acta Math. (IF 2.458) Pub Date : 2020-09-01
    Hélène Esnault; Michael Groechenig

    An irreducible integrable connection $(E,\nabla)$ on a smooth projective complex variety $X$ is called rigid if it gives rise to an isolated point of the corresponding moduli space $\mathcal{M}_{\rm dR}(X)$. According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss–Manin connection of a family of smooth projective

  • Small cancellation labellings of some infinite graphs and applications
    Acta Math. (IF 2.458) Pub Date : 2020-09-01
    Damian Osajda

    We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property

  • Stable solutions to semilinear elliptic equations are smooth up to dimension $9$
    Acta Math. (IF 2.458) Pub Date : 2020-06-23
    Xavier Cabré; Alessio Figalli; Xavier Ros-Oton; Joaquim Serra

    In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension $n \leqslant 9$. This result, that was only known to be true for $n \leqslant 4$, is optimal: $\log (1 / {\lvert x \rvert}^2)$ is a $W^{1,2}$ singular stable solution for $n \geqslant 10$. The proof of this conjecture is a consequence of a new

  • Non-collision singularities in a planar 4-body problem
    Acta Math. (IF 2.458) Pub Date : 2020-06-23
    Jinxin Xue

    In this paper, we show that there is a Cantor set of initial conditions in the planar $4$‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the $4$‑body case, and thus settles the last open case of the conjecture.

  • Local Hodge theory of Soergel bimodules
    Acta Math. (IF 2.458) Pub Date : 2017-07-07
    Geordie Williamson

    We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.

  • Maximum independent sets on random regular graphs
    Acta Math. (IF 2.458) Pub Date : 2017-07-07
    Jian Ding,Allan Sly,Nike Sun

    We determine the asymptotics of the independence number of the random d -regular graph for all \({d\geq d_0}\). It is highly concentrated, with constant-order fluctuations around \({n\alpha_*-c_*\log n}\) for explicit constants \({\alpha_*(d)}\) and \({c_*(d)}\). Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more

  • Global bifurcation of steady gravity water waves with critical layers
    Acta Math. (IF 2.458) Pub Date : 2017-07-07
    Adrian Constantin,Walter Strauss,Eugen Vărvărucă

    We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural

  • Lower bounds for numbers of real solutions in problems of Schubert calculus
    Acta Math. (IF 2.458) Pub Date : 2017-02-17
    Evgeny Mukhin,Vitaly Tarasov

    We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian \({\mathop{\rm Gr}(n,d)}\) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to \({\mathop{\rm gl}_n}\). The Gaudin Hamiltonians are self-adjoint with respect to a non-degenerate indefinite Hermitian form. Our bound

  • Universality in several-matrix models via approximate transport maps
    Acta Math. (IF 2.458) Pub Date : 2017-02-17
    Alessio Figalli,Alice Guionnet

    We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices (i.e., they are given by the sine-kernel in the bulk and the Tracy–Widom distribution at the edge), and we show averaged energy universality (i.e., universality for averages of m -points correlation

  • Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders
    Acta Math. (IF 2.458) Pub Date : 2017-02-17
    Patrick Bernard,Vadim Kaloshin,Ke Zhang

    We prove a form of Arnold diffusion in the a-priori stable case. Let $$H_{0}(p)+\epsilon H_{1}(\theta,p,t),\quad \theta \in {\mathbb{T}^{n}},\,p \in B^{n},\,t \in \mathbb{T}= \mathbb{R}/\mathbb{T},$$be a nearly integrable system of arbitrary degrees of freedom \({n \geqslant 2}\) with a strictly convex H 0. We show that for a “generic” \({\epsilon H_1}\), there exists an orbit \({(\theta,p)}\) satisfying

  • Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems
    Acta Math. (IF 2.458) Pub Date : 2016-11-16
    Yong Huang,Erwin Lutwak,Deane Yang,Gaoyong Zhang

    A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration

  • Helicoidal minimal surfaces of prescribed genus
    Acta Math. (IF 2.458) Pub Date : 2016-11-16
    David Hoffman,Martin Traizet,Brian White

    For every genus g , we prove that \({\mathbf{S}^2\times\mathbf{R}}\) contains complete, properly embedded, genus- g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the \({\mathbf{S}^2}\) tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in \({\mathbf{R}^3}\) that are helicoidal

  • A complete classification of homogeneous plane continua
    Acta Math. (IF 2.458) Pub Date : 2016-11-16
    Logan C. Hoehn,Lex G. Oversteegen

    We show that the only compact and connected subsets (i.e. continua ) X of the plane \({\mathbb{R}^2}\) which contain more than one point and are homogeneous, in the sense that the group of homeomorphisms of X acts transitively on X , are, up to homeomorphism, the circle \({\mathbb{S}^1}\), the pseudo-arc, and the circle of pseudo-arcs. These latter two spaces are fractal-like objects which do not contain

  • Regularity of Kähler–Ricci flows on Fano manifolds
    Acta Math. (IF 2.458) Pub Date : 2016-06-02
    Gang Tian,Zhenlei Zhang

    In this paper, we will establish a regularity theory for the Kähler–Ricci flow on Fano n -manifolds with Ricci curvature bounded in L p -norm for some \({p > n}\). Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau–Tian–Donaldson conjecture for Fano 3-manifolds. The results have been announced in [45].

  • The Hodge conjecture and arithmetic quotients of complex balls
    Acta Math. (IF 2.458) Pub Date : 2016-06-02
    Nicolas Bergeron,John Millson,Colette Moeglin

    Let S be a closed Shimura variety uniformized by the complex n -ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in \({H^{2k} (S, \mathbb{Q})}\), \({k=0,\dots, n}\), is algebraic. We show that this holds for all degrees k away from the neighborhood \({\bigl]\tfrac13n,\tfrac23n\bigr[}\) of the middle degree. We also prove the Tate conjecture for the

  • New partially hyperbolic dynamical systems I
    Acta Math. (IF 2.458) Pub Date : 2016-02-01
    Andrey Gogolev,Pedro Ontaneda,Federico Rodriguez Hertz

    We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms

  • Existence and classification of overtwisted contact structures in all dimensions
    Acta Math. (IF 2.458) Pub Date : 2016-02-01
    Matthew Strom Borman,Yakov Eliashberg,Emmy Murphy

    We establish a parametric extension h -principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from [12]. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.

  • Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials
    Acta Math. (IF 2.458) Pub Date : 2016-02-01
    Alexander I. Aptekarev,Maxim L. Yattselev

    Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, \({f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}\), \({\# A< \infty}\). J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal

  • Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture
    Acta Math. (IF 2.458) Pub Date : 2016-02-01
    Pramod N. Achar,Laura Rider

    We prove the Mirković–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p -torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about

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