Showing papers in "Geometric and Functional Analysis in 2016"

Kähler–Einstein metrics along the smooth continuity method

Abstract: We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kahler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kahler–Einstein manifolds. We also give analogous results for twisted Kahler–Einstein metrics and Kahler–Ricci solitons.

KAM for the nonlinear beam equation

Abstract: In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0, \quad t \in {\mathbb{R}}, x \in {\mathbb{T}^d}, \quad (*)$$ where \({G(x,u)=u^4+ O(u^5)}\). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation \({(*)_{G=0}}\), written as the system $$u_t=-v,\quad v_t=\Delta^2 u+mu,$$ persist as invariant tori of the nonlinear equation \({(*)}\), re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of \({(*)}\). If \({d\ge2}\), then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way.

Spectral gaps, additive energy, and a fractal uncertainty principle

Abstract: We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension $${\delta}$$ of the limit set close to $${{n-1\over 2}}$$ . The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors–David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Abstract: “Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincare 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Ito chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

No-gaps delocalization for general random matrices

Abstract: We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its $${\ell_2}$$ norm. Our results pertain to a wide class of random matrices, including matrices with independent entries, symmetric and skew-symmetric matrices, as well as some other naturally arising ensembles. The matrices can be real and complex; in the latter case we assume that the real and imaginary parts of the entries are independent.

From volume cone to metric cone in the nonsmooth setting

Abstract: We prove that ‘volume cone implies metric cone’ in the setting of $${\mathsf{RCD}}$$ spaces, thus generalising to this class of spaces a well known result of Cheeger–Colding valid in Ricci-limit spaces.

Approximating C 0-foliations by contact structures

Abstract: We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C 0-approximated by both positive and negative contact structures unless all leaves of the foliation are simply connected. As applications we deduce that the existence of a taut C 0-foliation implies the existence of universally tight contact structures in the same homotopy class of plane fields and that a closed 3-manifold that admits a taut C 0-foliation of codimension-1 is not an L-space in the sense of Heegaard-Floer homology.

Modular curvature and Morita equivalence

Abstract: We prove that the modular curvature of a conformal metric structure on the noncommutative torus T 2 θ (θ / 2 Q) is invariant under Morita equivalence. More precisely, the curvature associated to a Hermitian structure on a Heisenberg bimodule E realizing the Morita equivalence between Aθ = C(T 2 θ) and Aθ', with Aθ identified to the algebra of endomorphisms EndA�'(E), coincides with the intrinsic curvature of the conformal metric on T 2 θ with corresponding Weyl factor. The main analytical tool is the extension of Connes' pseudodifferential calculus to Heisenberg modules, the novel technical aspect being that the entire computation is free of any computer assistance.

Rectifiability of harmonic measure

Abstract: In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).

Mean dimension of $\mathbb{Z}^k$-actions

Abstract: Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension. (1) When is X isomorphic to the inverse limit of finite entropy systems? (2) Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer? (3) When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

Abstract: For every finite measure \({\mu}\) on \({{\mathbb{R}}^n}\) we define a decomposability bundle \({V(\mu,\,\cdot)}\) related to the decompositions of \({\mu}\) in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on \({{\mathbb{R}}^n}\) is differentiable at \({\mu}\)-a.e. \({x}\) with respect to the subspace \({V(\mu,\,x)}\), and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at \({\mu}\)-a.e. \({x}\) in any direction which is not in \({V(\mu,\,x)}\). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) \({k}\)-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class \({C^1}\) to Lipschitz maps.

Counting curves in hyperbolic surfaces

Abstract: Let \({\Sigma}\) be a hyperbolic surface. We study the set of curves on \({\Sigma}\) of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary \({\gamma_0}\). For example, in the particular case that \({\Sigma}\) is a once-punctured torus, we prove that the cardinality of the set of curves of type \({\gamma_0}\) and of at most length L is asymptotic to \({L^2}\) times a constant.

Finiteness Principles for Smooth Selection

Abstract: In this paper we prove finiteness principles for $${C^m{({\mathbb{R}^n},{\mathbb{R}^D)}}}$$ and $${C^{m-1,1}(\mathbb{R}^n,\mathbb{R}^D)}$$ selections. In particular, we provide a proof for a conjecture of Brudnyi-Shvartsman (1994) on Lipschitz selections for the case when the domain is $${X=\mathbb{R}^n}$$ .

Smoothing out positively curved metric cones by Ricci expanders

Abstract: We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expanders is connected.

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

Abstract: Expander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439–562, 2006; Lubotzky, Bull Am Math Soc, 49:113–162, 2012). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov (Geom Funct Anal 20(2):416–526, 2010), is whether bounded degree high dimensional expanders exist for \({d \geq 2}\). We present an explicit construction of bounded degree complexes of dimension \({d = 2}\) which are topological expanders, thus answering Gromov’s question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on \({\mathbb{F}_2}\) systolic invariants of these complexes, which seem to be the first linear\({\mathbb{F}_2}\) systolic bounds. The expansion results are deduced from these isoperimetric inequalities.

Apollonian structure in the Abelian sandpile

Abstract: The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.

Cutoff on all Ramanujan graphs

Abstract: We show that on every Ramanujan graph $${G}$$ , the simple random walk exhibits cutoff: when $${G}$$ has $${n}$$ vertices and degree $${d}$$ , the total-variation distance of the walk from the uniform distribution at time $${t=\frac{d}{d-2} \log_{d-1} n + s\sqrt{\log n}}$$ is asymptotically $${{\mathbb{P}}(Z > c \, s)}$$ where $${Z}$$ is a standard normal variable and $${c=c(d)}$$ is an explicit constant. Furthermore, for all $${1 \leq p \leq \infty}$$ , $${d}$$ -regular Ramanujan graphs minimize the asymptotic $${L^p}$$ -mixing time for SRW among all $${d}$$ -regular graphs. Our proof also shows that, for every vertex $${x}$$ in $${G}$$ as above, its distance from $${n-o(n)}$$ of the vertices is asymptotically $${\log_{d-1} n}$$ .

Maximizers for the Stein–Tomas Inequality

Abstract: We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension.

Spectral multipliers on 2-step groups: topological versus homogeneous dimension

Abstract: Let G be a 2-step stratified group of topological dimension d and homogeneous dimension Q. Let $${\mathcal{L}}$$ be a homogeneous sub-Laplacian on G. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $${F(\mathcal{L})}$$ is of weak type (1, 1) and bounded on L p (G) for all p ∈ (1, ∞) whenever the multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2. It is known that, for several 2-step groups and sub-Laplacians, the threshold Q/2 in the smoothness condition is not sharp and in many cases it is possible to push it down to d/2. Here we show that, for all 2-step groups and sub-Laplacians, the sharp threshold is strictly less than Q/2, but not less than d/2.

Fibrations with few rational points

Abstract: We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises and unifies existing results in the literature by Serre, Browning–Dietmann, Bright–Browning–Loughran, Graber–Harris–Mazur–Starr, et al.

On semiconjugate rational functions

Abstract: We investigate semiconjugate rational functions, that is rational functions A, B related by the functional equation \({A \circ X = X \circ B}\), where X is a rational function. We show that if A and B is a pair of such functions, then either A can be obtained from B by a certain iterative process, or A and B can be described in terms of orbifolds of non-negative Euler characteristic on the Riemann sphere.

A {C^\infty} closing lemma for Hamiltonian diffeomorphisms of closed surfaces

Abstract: We prove a $${C^\infty}$$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $${C^\infty}$$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.

W*-rigidity for the von Neumann algebras of products of hyperbolic groups

Abstract: We show that if \({\Gamma = \Gamma_1\times\dotsb\times \Gamma_n}\) is a product of \({{\rm n} \geq 2}\) non-elementary ICC hyperbolic groups then any discrete group \({\Lambda}\) which is \({W^*}\)-equivalent to \({\Gamma}\) decomposes as a direct product of n ICC groups and does not decompose as a direct product of k ICC groups when \({{\rm n} ot= {\rm k}}\). This gives a group-level strengthening of Ozawa and Popa’s unique prime decomposition theorem by removing all assumptions on the group \({\Lambda}\). This result in combination with Margulis’ normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II1 factors.

Fuglede–Kadison Determinants and Sofic Entropy

Abstract: We relate Fuglede–Kadison determinants to entropy of finitely-presented algebraic actions in essentially complete generality. We show that if \({f\in M_{m,n}(\mathbb{Z}(\Gamma))}\) is injective as a left multiplication operator on \({\ell^{2}(\Gamma)^{\oplus n},}\) then the topological entropy of the action of \({\Gamma}\) on the dual of \({\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus m}f}\) is at most the logarithm of the positive Fuglede–Kadison determinant of f, with equality if m = n. We also prove that when m = n the measure-theoretic entropy of the action of \({\Gamma}\) on the dual of \({\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus n}f}\) is the logarithm of the Fuglede–Kadison determinant of f. This work completely settles the connection between entropy of principal algebraic actions and Fuglede–Kadison determinants in the generality in which dynamical entropy is defined. Our main Theorem partially generalizes results of Li-Thom from amenable groups to sofic groups. Moreover, we show that the obvious full generalization of the Li-Thom theorem for amenable groups is false for general sofic groups. Lastly, we undertake a study of when the Yuzvinskiǐ addition formula fails for a non-amenable sofic group \({\Gamma}\), showing it always fails if \({\Gamma}\) contains a nonabelian free group, and relating it to the possible values of L2-torsion in general.

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

Abstract: We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.

Convexity and Zariski decomposition structure

Abstract: This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre–Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.

Arithmetic Groups, Base Change, and Representation Growth

Abstract: Consider an arithmetic group $${\mathbf{G}(O_S)}$$ , where $${\mathbf{G}}$$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers O S of a number field K with respect to a finite set of places S. For each $${n \in \mathbb{N}}$$ , let $${R_n(\mathbf{G}(O_S))}$$ denote the number of irreducible complex representations of $${\mathbf{G}(O_S)}$$ of dimension at most n. The degree of representation growth $${\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty} \log R_n(\mathbf{G}(O_S)) / \log n}$$ is finite if and only if $${\mathbf{G}(O_S)}$$ has the weak Congruence Subgroup Property. We establish that for every $${\mathbf{G}(O_S)}$$ with the weak Congruence Subgroup Property the invariant $${\alpha(\mathbf{G}(O_S))}$$ is already determined by the absolute root system of $${\mathbf{G}}$$ . To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions $${K{\subset}L}$$ . We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines.

Effective equidistribution of twisted horocycle flows and horocycle maps

Abstract: We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of Venkatesh, recently already improved by Tanis and Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by Shah and Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cusp forms which are comparable to the best known bounds of Good in the holomorphic case, and of Bernstein and Reznikov in the Maass (non-holomorphic) case. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.

Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

Abstract: We show that if \({f\colon X\to Y}\) is a quasisymmetric mapping between Ahlfors regular spaces, then \({dim_H f(E)\leq dim_H E}\) for “almost every” bounded Ahlfors regular set \({E\subseteq X}\). If additionally, \({X}\) and \({Y}\) are Loewner spaces then \({dim_H f(E)=dim_H E}\) for “almost every" Ahlfors regular set \({E\subset X}\). The precise statements of these results are given in terms of Fuglede’s modulus of measures. As a corollary of these general theorems we show that if \({f}\) is a quasiconformal map of \({\mathbb{R}^N}\), \({N\geq 2}\), then for Lebesgue a.e. \({y\in\mathbb{R}^N}\) we have \({dim_H f(y+E) = dim_H E}\). A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if \({E \subset {\mathbb{R}}}\) is Ahlfors \({d}\)-regular, \({d < 1}\), then some component of \({f(E \times {\mathbb{R}})}\) has dimension at most \({2/(d+1)}\), and we construct examples to show this bound is sharp. In addition, we show there is a \({1}\)-dimensional set \({S\subseteq \mathbb R}\) and planar quasiconformal map \({f}\) such that \({f({\mathbb{R}} \times S)}\) contains no rectifiable sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric, 2016).

On quasihomomorphisms with noncommutative targets

Abstract: We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are “constructible”, i.e., are obtained via certain natural operations from homomorphisms to some groups and quasihomomorphisms to abelian groups. We illustrate this theorem by describing quasihomomorphisms to certain classes of groups. For instance, every unbounded quasihomomorphism to a torsion-free hyperbolic group H is either a homomorphism to a subgroup of H or is a quasihomomorphism to an infinite cyclic subgroup of H.