Authors:Logan C. Hoehn; Lex G. Oversteegen Pages: 177 - 216 Abstract: 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 any arcs. It follows that any compact and homogeneous space in the plane has the form X × Z, where X is either a point or one of the three homogeneous continua above, and Z is either a finite set or the Cantor set. The main technical result in this paper is a new characterization of the pseudo-arc. Following Lelek, we say that a continuum X has span zero provided for every continuum C and every pair of maps \({f,g\colon C \to X}\) such that \({f(C) \subset g(C)}\) there exists \({c_0 \in C}\) so that f(c 0) = g(c 0). We show that a continuum is homeomorphic to the pseudo-arc if and only if it is hereditarily indecomposable (i.e., every subcontinuum is indecomposable) and has span zero. PubDate: 2016-06-01 DOI: 10.1007/s11511-016-0138-0 Issue No:Vol. 216, No. 2 (2016)

Authors:David Hoffman; Martin Traizet; Brian White Pages: 217 - 323 Abstract: 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 at infinity. We prove that helicoidal surfaces in \({\mathbf{R}^3}\) of every prescribed genus occur as such limits of examples in \({\mathbf{S}^2\times\mathbf{R}}\) . PubDate: 2016-06-01 DOI: 10.1007/s11511-016-0139-z Issue No:Vol. 216, No. 2 (2016)

Authors:Yong Huang; Erwin Lutwak; Deane Yang; Gaoyong Zhang Pages: 325 - 388 Abstract: 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, are established for the existence of solutions to these problems. PubDate: 2016-06-01 DOI: 10.1007/s11511-016-0140-6 Issue No:Vol. 216, No. 2 (2016)

Authors:Nicolas Bergeron; John Millson; Colette Moeglin Pages: 1 - 125 Abstract: 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 same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups \({GL (N) \rtimes \langle \theta \rangle}\) associated with base change. PubDate: 2016-03-01 DOI: 10.1007/s11511-016-0136-2 Issue No:Vol. 216, No. 1 (2016)

Authors:Gang Tian; Zhenlei Zhang Pages: 127 - 176 Abstract: 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]. PubDate: 2016-03-01 DOI: 10.1007/s11511-016-0137-1 Issue No:Vol. 216, No. 1 (2016)

Abstract: 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. PubDate: 2016-12-01 DOI: 10.1007/s11511-017-0146-8

Abstract: 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 broadly applicable to the study of other combinatorial properties of random graphs. PubDate: 2016-12-01 DOI: 10.1007/s11511-017-0145-9

Abstract: 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 properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation. PubDate: 2016-12-01 DOI: 10.1007/s11511-017-0144-x

Abstract: 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 comes from the computation of the signature of that form. PubDate: 2016-09-01 DOI: 10.1007/s11511-016-0143-3

Abstract: 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 functions around some energy level E in the bulk). As a corollary, these results yield universality for self-adjoint polynomials in several independent GUE or GOE matrices which are close to the identity. PubDate: 2016-09-01 DOI: 10.1007/s11511-016-0142-4

Abstract: 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 $$\ p(t)-p(0)\ > l(H_{1}) > 0,$$ where \({l(H_1)}\) is independent of \({\epsilon}\) . The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case. PubDate: 2016-09-01 DOI: 10.1007/s11511-016-0141-5

Authors:Pramod N. Achar; Laura Rider Pages: 183 - 216 Abstract: 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 perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves. PubDate: 2015-12-01 DOI: 10.1007/s11511-016-0132-6 Issue No:Vol. 215, No. 2 (2015)

Authors:Alexander I. Aptekarev; Maxim L. Yattselev Pages: 217 - 280 Abstract: 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 Padé approximants for f. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function \({f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}\) . The complete proof of Nuttall’s conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro-logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points. PubDate: 2015-12-01 DOI: 10.1007/s11511-016-0133-5 Issue No:Vol. 215, No. 2 (2015)

Authors:Matthew Strom Borman; Yakov Eliashberg; Emmy Murphy Pages: 281 - 361 Abstract: 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. PubDate: 2015-12-01 DOI: 10.1007/s11511-016-0134-4 Issue No:Vol. 215, No. 2 (2015)

Authors:Andrey Gogolev; Pedro Ontaneda; Federico Rodriguez Hertz Pages: 363 - 393 Abstract: 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 in the past forty years. PubDate: 2015-12-01 DOI: 10.1007/s11511-016-0135-3 Issue No:Vol. 215, No. 2 (2015)

Authors:Artur Avila Pages: 1 - 54 Abstract: We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large. Key to our approach are two results about the dependence of the Lyapunov exponent of one-frequency SL \({(2,\mathbb{C})}\) cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe “quantization” restriction, and on the other hand “regularity” of the dependence characterizes uniform hyperbolicity when the Lyapunov exponent is positive. Our method is independent of arithmetic conditions on the frequency. PubDate: 2015-09-01 DOI: 10.1007/s11511-015-0128-7 Issue No:Vol. 215, No. 1 (2015)

Authors:David Dumas Pages: 55 - 126 Abstract: We show that Thurston’s skinning maps of Teichmüller space have finite fibers. The proof centers around a study of two subvarieties of the \({{\rm SL}_2(\mathbb{C})}\) character variety of a surface—one associated with complex projective structures, and the other associated with a 3-manifold. Using the Morgan–Shalen compactification of the character variety and author’s results on holonomy limits of complex projective structures, we show that these subvarieties have only a discrete set of intersections. Along the way, we introduce a natural stratified Kähler metric on the space of holomorphic quadratic differentials on a Riemann surface and show that it is symplectomorphic to the space of measured foliations. Mirzakhani has used this symplectomorphism to show that the Hubbard–Masur function is constant; we include a proof of this result. We also generalize Floyd’s theorem on the space of boundary curves of incompressible, boundary-incompressible surfaces to a statement about extending group actions on \({\Lambda}\) -trees. PubDate: 2015-09-01 DOI: 10.1007/s11511-015-0129-6 Issue No:Vol. 215, No. 1 (2015)

Authors:Alexander Gorodnik; Ralf Spatzier Pages: 127 - 159 Abstract: We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms. PubDate: 2015-09-01 DOI: 10.1007/s11511-015-0130-0 Issue No:Vol. 215, No. 1 (2015)

Authors:Misha Verbitsky Pages: 161 - 182 Abstract: Let M be a compact complex manifold. The corresponding Teichmüller space Teich is the space of all complex structures on M up to the action of the group \({{\rm Diff}_0(M)}\) of isotopies. The mapping class group \({\Gamma:={\rm Diff}(M)/{{\rm Diff}_0(M)}}\) acts on Teich in a natural way. An ergodic complex structure is a complex structure with a \({\Gamma}\) -orbit dense in Teich. Let M be a complex torus of complex dimension \({\ge 2}\) or a hyperkähler manifold with \({b_2 > 3}\) . We prove that M is ergodic, unless M has maximal Picard rank (there are countably many such M). This is used to show that all hyperkähler manifolds are Kobayashi non-hyperbolic. PubDate: 2015-09-01 DOI: 10.1007/s11511-015-0131-z Issue No:Vol. 215, No. 1 (2015)

Authors:Andrea Pulita Pages: 307 - 355 Abstract: We prove that the radii of convergence of the solutions of a p-adic differential equation \({\fancyscript{F}}\) over an affinoid domain X of the Berkovich affine line are continuous functions on X that factorize through the retraction of \({X\to\Gamma}\) of X onto a finite graph \({\Gamma\subseteq X}\) . We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on X is controlled by a finite family of data. PubDate: 2015-06-01 DOI: 10.1007/s11511-015-0126-9 Issue No:Vol. 214, No. 2 (2015)