Showing papers in "Geometric and Functional Analysis in 2019"

An L2-identity and pinned distance problem

Abstract: Let $${\mu}$$ be a Frostman measure on $${E\subset\mathbb{R}^d}$$ . The spherical average estimate $$\int_{S^{d-1}}|\widehat{\mu}(r\omega)|^2\,d\omega\lesssim r^{-\beta}$$ was originally used to attack Falconer distance conjecture, via Mattila’s integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average estimates imply the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres–Schlag’s result on pinned distance problem significantly. The idea in our approach is to reduce the pinned distance problem to an integral where spherical averages apply. The key new ingredient is the following identity. Using a group action argument, we show that for any Schwartz function f on $${\mathbb{R}^d}$$ and any $${x\in\mathbb{R}^d}$$ , $$\int_0^\infty|\omega_t*f(x)|^2\,t^{d-1}dt\,=\int_0^\infty|\widehat{\omega_r}*f(x)|^2\,r^{d-1}dr,$$ where $${\omega_r}$$ is the normalized surface measure on $${r S^{d-1}}$$ . An interesting remark is that the right hand side can be easily seen equal to $$c_d\int\left|D_x^{-\frac{d-1}{2}}e^{-2\pi it\sqrt{-\Delta}}f(x)\right|^2\,dt=c_d'\int\left|D_x^{-\frac{d-2}{2}}e^{2\pi{it}\Delta}f(x)\right|^2\,dt.$$ An alternative derivation of Mattila’s integral via group actions is also given in the Appendix.

A steady Euler flow with compact support

Abstract: A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

Abstract: We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

The Weyl Law for the phase transition spectrum and density of limit interfaces

Abstract: We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich–Marques–Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie–Marques–Neves and Marques–Neves–Song, respectively. We also prove the density of separating limit interfaces for generic metrics in dimension 3, based on the recent work of Chodosh–Mantoulidis, and for generic metrics on manifolds containing only separating minimal hypersurfaces, e.g. $${H_{n}(M,\mathbb{Z}_2) = 0}$$ , for $${4 \leq n + 1 \leq 7}$$ . These provide alternative proofs of Yau’s conjecture on the existence of infinitely many minimal hypersurfaces for generic metrics on each setting, using the Allen–Cahn approach.

Geodesically complete spaces with an upper curvature bound

Abstract: We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

Polynomial bound for partition rank in terms of analytic rank

Abstract: Let $$G_1, \ldots , G_k$$ be vector spaces over a finite field $${\mathbb {F}} = {\mathbb {F}}_q$$ with a non-trivial additive character $$\chi $$ . The analytic rank of a multilinear form $$\alpha :G_1 \times \cdots \times G_k \rightarrow {\mathbb {F}}$$ is defined as $${\text {arank}}(\alpha ) = -\log _q \mathop {\mathbb {E}} _{x_1 \in G_1, \ldots , x_k\in G_k} \chi (\alpha (x_1,\ldots , x_k))$$ . The partition rank $${\text {prank}}(\alpha )$$ of $$\alpha $$ is the smallest number of maps of partition rank 1 that add up to $$\alpha $$ , where a map is of partition rank 1 if it can be written as a product of two multilinear forms, depending on different coordinates. It is easy to see that $${\text {arank}}(\alpha ) \le O({\text {prank}}(\alpha ))$$ and it has been known that $${\text {prank}}(\alpha )$$ can be bounded from above in terms of $${\text {arank}}(\alpha )$$ . In this paper, we improve the latter bound to polynomial, i.e. we show that there are quantities C, D depending on k only such that $${\text {prank}}(\alpha ) \le C ({\text {arank}}(\alpha )^D + 1)$$ . As a consequence, we prove a conjecture of Kazhdan and Ziegler. The same result was obtained independently and simultaneously by Janzer.

Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications

Abstract: We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]).

Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components

Abstract: In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical surfaces with conical points to the associated moduli space of pointed Riemann surfaces, such as its properness, which follows from an explicit systole inequality that relates metric invariants (spherical systole) and conformal invariant (extremal systole).

The base change in the Atiyah and the Lück approximation conjectures

Abstract: Let F be a free finitely generated group and $${A \in {\rm Mat}_{n \times m}(\mathbb{C}[F])}$$ . For each quotient G = F/N of F we can define a von Neumann rank function rkG(A) associated with the l2-operator l2(G)n → l2(G)m induced by right multiplication by A. For example, in the case where G is finite, $${{\rm rk}_G(A)=\frac{{\rm rk}_{\mathbb{C}}({{\bar{A}}})}{|G|}}$$ is the normalized rank of the matrix $${{{\bar{A}}} \in {\rm Mat}_{n \times m}(\mathbb{C}[G])}$$ obtained by reducing the coefficients of A modulo N. One of the variations of the Luck approximation conjecture claims that the function $${N\mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of G is finite, then $${{\rm rk}_G(A) \in \frac{1}{lcm (G)}\mathbb{Z}}$$ . In our first result we prove the sofic Luck approximation conjecture. In particular, we show that the function $${N \mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Luck approximation and we show that the strong Atiyah conjecture holds for groups from a class $${{\mathcal{D}}}$$ , virtually compact special groups, Artin’s braid groups and torsion-free p-adic analytic pro-p groups.

Double variational principle for mean dimension

Abstract: We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

Percolation on Hyperbolic Graphs

Abstract: We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-transitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that $$p_c

Long-time existence for multi-dimensional periodic water waves

Abstract: We prove an extended lifespan result for the full gravity-capillary water waves system with a 2 dimensional periodic interface: for initial data of sufficiently small size $${\varepsilon}$$ , smooth solutions exist up to times of the order of $${\varepsilon^{-5/3+}}$$ , for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of the equations which gives additional smoothing close to the resonant hypersurfaces, (2) another structural property, connected to time-reversibility, that allows us to handle “trivial” cubic resonances, (3) sharp small divisors lower bounds on three and four-way modulation functions based on counting arguments, and (4) partial normal form transformations and symmetrization arguments in the Fourier space. Our theorem appears to be the first extended lifespan result for quasilinear equations with non-trivial resonances on a multi-dimensional torus.

Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

Abstract: Given a non-conformal repeller $$\Lambda $$ of a $$C^{1+\gamma }$$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok’s approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure.

Application of signal analysis to the embedding problem of $\mathbb{Z}^k$-actions

Abstract: We study the problem of embedding arbitrary $${\mathbb {Z}}^k$$ -actions into the shift action on the infinite dimensional cube $$\left( [0,1]^D\right) ^{{\mathbb {Z}}^k}$$ . We prove that if a $${\mathbb {Z}}^k$$ -action X satisfies the marker property (in particular if X is a minimal system without periodic points) and if its mean dimension is smaller than D / 2 then we can embed it in the shift on $$\left( [0,1]^D\right) ^{{\mathbb {Z}}^k}$$ . The value D / 2 here is optimal. The proof goes through signal analysis. We develop the theory of encoding $${\mathbb {Z}}^k$$ -actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of $${\mathbb {R}}^k$$ into a band-limited function which is constructed from another tiling.

Anderson localization for two interacting quasiperiodic particles

Abstract: We consider a system of two discrete quasiperiodic 1D particles as an operator on $$\ell^2(\mathbb Z^2)$$ and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to those values.

Rigidity of the 1-Bakry–Émery Inequality and Sets of Finite Perimeter in RCD Spaces

Abstract: This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over $${{\,\mathrm{RCD}\,}}(K,N)$$ metric measure spaces. Our main result asserts existence of a Euclidean tangent half-space almost everywhere with respect to the perimeter measure and it can be improved to an existence and uniqueness statement when the ambient is non collapsed. As an intermediate tool, we provide a complete characterization of the class of $${{\,\mathrm{RCD}\,}}(0,N)$$ spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry–Emery inequality. This result is of independent interest and it is new, up to our knowledge, even in the smooth framework.

Finitary random interlacements and the Gaboriau–Lyons problem

Abstract: The von Neumann–Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol’shanskii in the 1980s. The measurable version (formulated by Gaboriau–Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau–Lyons. The proof uses an approximation to the random interlacement process by random multisets of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau–Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.

Geodesic Currents and Counting Problems

Abstract: For every positive, continuous and homogeneous function f on the space of currents on a compact surface $${{{\overline{\Sigma }}}}$$ , and for every compactly supported filling current $$\alpha $$ , we compute as $$L \rightarrow \infty $$ , the number of mapping classes $$\phi $$ so that $$f(\phi (\alpha ))\le L$$ . As an application, when the surface in question is closed, we prove a lattice counting theorem for Teichmuller space equipped with the Thurston metric.

Incidence Estimates for Well Spaced Tubes

Abstract: We prove analogues of the Szemeredi–Trotter theorem and other incidence theorems using $$\delta $$-tubes in place of straight lines, assuming that the $$\delta $$-tubes are well spaced in a strong sense.

Density of algebraic points on Noetherian varieties

Abstract: Let $${\Omega \subset \mathbb{R}^{n}}$$ be a relatively compact domain. A finite collection of real valued functions on $${{\Omega}}$$ is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noetherian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila–Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any $${\varepsilon > 0}$$ , the number of points of height H in the transcendental part of the set is at most C· $${{H}^ \varepsilon}$$ where C can be explicitly estimated from the Noetherian parameters and $${\varepsilon}$$ . We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety $${\mathcal{A}_{g}}$$ . We thus effectivize the (geometric side of) Pila–Zannier strategy for unlikely intersections in various contexts.

Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree

Abstract: In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials, is obtained by reducing the problem to the construction of a particular example with strong hyperbolicity properties. This approach relies the construction of higher order logarithmic connections allowing us to construct logarithmic Wronskians. These logarithmic Wronskians are the building blocks of the more general logarithmic jet differentials we are able to construct. As a byproduct of our proof, we prove a more general result on the orbifold hyperbolicity for generic geometric orbifolds in the sense of Campana, with only one component and large multiplicities. We also establish a Second Main Theorem type result for holomorphic entire curves intersecting general hypersurfaces, and we prove the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface, again with an explicit lower bound on the degree of all these hypersurfaces.

On the Yang–Yau inequality for the first Laplace eigenvalue

Abstract: In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus $$\gamma $$ and the area. The equality in Yang–Yau’s estimate is attained for $$\gamma =0$$ by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for $$\gamma =2$$. In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang–Yau’s inequality is strict for all genera $$\gamma >2$$. Previously this was only known for $$\gamma =1$$. In the second part of the paper we apply Chern-Wolfson’s notion of harmonic sequence to obtain an upper bound on the total branching order of harmonic maps from surfaces to spheres. Applications of these results to extremal metrics for eigenvalues are discussed.

Large Genus Asymptotics for Siegel–Veech Constants

Abstract: In this paper we consider the large genus asymptotics for two classes of Siegel–Veech constants associated with an arbitrary connected stratum $$\mathcal {H} (\alpha )$$ of Abelian differentials. The first is the saddle connection Siegel–Veech constant $$c_{\mathrm{sc}}^{m_i, m_j} \big ( \mathcal {H} (\alpha ) \big )$$ counting saddle connections between two distinct, fixed zeros of prescribed orders $$m_i$$ and $$m_j$$ , and the second is the area Siegel–Veech constant $$c_{\mathrm{area}} \big ( \mathcal {H}(\alpha ) \big )$$ counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of Eskin–Masur–Zorich that express these constants in terms of Masur–Veech strata volumes, with a recent result for the large genus asymptotics of these volumes, we show that $$c_{\mathrm{sc}}^{m_i, m_j} \big ( \mathcal {H} (\alpha ) \big ) = (m_i + 1) (m_j + 1) \big ( 1 + o(1) \big )$$ and $$c_{\mathrm{area}} \big ( \mathcal {H}(\alpha ) \big ) = \frac{1}{2} + o(1)$$ , both as $$|\alpha | = 2g - 2$$ tends to $$\infty $$ . The former result confirms a prediction of Zorich and the latter confirms one of Eskin–Zorich in the case of connected strata.

A gluing construction of collapsing Calabi–Yau metrics on K3 fibred 3-folds

Abstract: We use the gluing method to give a refined description of the collapsing Calabi–Yau metrics on Calabi–Yau 3-folds admitting a Lefschetz K3 fibration.

Affine actions with Hitchin linear part

Abstract: Properly discontinuous actions of a surface group by affine automorphisms of $${\mathbb {R}}^d$$ were shown to exist by Danciger–Gueritaud–Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in $${{\mathsf {S}}}{{\mathsf {O}}}(n,n-1)$$ , so that the affine action is by isometries of a flat pseudo-Riemannian metric on $${\mathbb {R}}^d$$ of signature $$(n,n-1)$$ . Here, the translational part determines a deformation of the linear part into $$\mathsf {PSO}(n,n)$$ -Hitchin representations and the crucial step is to show that such representations are not Anosov in $$\mathsf {PSL}(2n,{\mathbb {R}})$$ with respect to the stabilizer of an n-plane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudo-Riemannian hyperbolic space of signature $$(n,n-1)$$ by a $$\mathsf {PSO}(n,n)$$ -Hitchin representation fails to be properly discontinuous.

Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

Abstract: We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local (1, p)-Poincare inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new “thickening” construction, which can be used to enlarge subsets into spaces admitting Poincare inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincare inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincare inequalities for the class of MCP(K, n)-spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincare inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincare inequalities “self-improve” to classical (1, q)-Poincare inequalities for some $${q \in [1,\infty)}$$ , which is related to Keith’s and Zhong’s theorem on self-improvement of Poincare inequalities.

Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

Abstract: We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

Riemannian curvature measures

Abstract: A famous theorem of Weyl states that if M is a compact submanifold of euclidean space, then the volumes of small tubes about M are given by a polynomial in the radius r, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of M with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of M. This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra $${\mathbb{R}[t]}$$ that reflects the behavior of Alesker multiplication. This module encodes a key piece of the array of kinematic formulas of any Riemannian manifold on which a group of isometries acts transitively on the sphere bundle. We illustrate this principle in precise terms in the case where M is a complex space form.

The sparse circular law under minimal assumptions

Abstract: The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $${n \times n}$$ matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension n grows to infinity. Consider an $${n \times n}$$ matrix $${A_n=(\delta_{ij}^{(n)}\xi_{ij}^{(n)})}$$ , where $${\xi_{ij}^{(n)}}$$ are copies of a real random variable of unit variance, variables $${\delta_{ij}^{(n)}}$$ are Bernoulli (0/1) with $${\mathbb{P}\{\delta_{ij}^{(n)} = 1\} = p_n}$$ , and $${\delta_{ij}^{(n)}}$$ and $${\xi_{ij}^{(n)}, i, j \in [n]}$$ , are jointly independent. In order for the circular law to hold for the sequence $${\big(\frac{1}{\sqrt{p_{n}n}}A_{n}\big)}$$ , one has to assume that $${p_{n}n \to \infty}$$ . We derive the circular law under this minimal assumption.

Small Gaps of GOE

Abstract: In this article, we study the smallest gaps between eigenvalues of the Gaussian orthogonal ensemble (GOE). The main result is that the smallest gaps, after being normalized by n, will converge to a Poisson distribution, and the limiting density of the kth normalized smallest gap is $$2{}x^{2k-1}e^{-x^{2}}/(k-1)!$$. The proof is based on the method developed in Feng and Wei (Small gaps of circular $$\beta $$-ensemble. arXiv:1806.01555). We need to prove the convergence of the factorial moments of the smallest gaps, which makes use of the Pfaffian structure of GOE and some comparison results between the one-component log-gas and the two-component log-gas.