Geometric and Functional Analysis最新文献

We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin. (3) The parallelizability of planar d-webs. (4) The Elekes-Rónyai theorem on expanding polynomials.

Given a Borel set A in the plane, we study the set of exceptional vantage points, for which the pinned distance Δ_p(A) has small dimension, that is, close to (dimA)/2. We show that if this set has positive dimension, then it must have very special structure. This result follows from a more general single-scale nonlinear projection theorem, which says that if ϕ₁, ϕ₂, ϕ₃ are three smooth functions whose associated 3-web has non-vanishing Blaschke curvature, and if A is a (δ,α)₂-set in the sense of Katz and Tao, then at least one of the images ϕ_i(A) must have measure much larger than |A|^1/2, where |A| stands for the measure of A. We prove analogous results for d smooth functions ϕ₁,…,ϕ_d, whose associated d-web is not parallelizable.

We use similar tools to characterize when bivariate real analytic functions are “dimension expanding” when applied to a Cartesian product: if P is a bivariate real analytic function, then P is either locally of the form h(a(x)+b(y)), or P(A,B) has dimension at least α+c whenever A and B are Borel sets with Hausdorff dimension α. Again, this follows from a single-scale estimate, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture.

Abstract

A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called Morse flats. We show that the Tits boundary ∂_TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂_TF. More precisely, we show that the set of singular points in ∂_TF can be covered by finitely many round spheres of positive codimension.

Abstract

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of (mathbb{R}^{n}) . We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.

Abstract

We give an optimal bound on normal curvatures of immersed n-torus in a Euclidean ball of large dimension.

We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson (Geom. Meas. Theor. Calcu. Var. 44:441, 1986, Problem 5.7).

The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in R⁴, while they do not exist in positively curved closed Riemannian (n+1)-manifold when n≤5; in particular, there are no stable minimal hypersurfaces in Sⁿ⁺¹ when n≤5. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other.

In this paper, we prove a formula, realizing certain residual Eisenstein series on symplectic groups as regularized kernel integrals. These Eisenstein series, as well as the kernel integrals, are attached to Speh representations. This forms an initial step to a new type of a regularized Siegel-Weil formula that we propose. This new formula bears the same relation to the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan, as does the regularized Siegel-Weil formula to the doubling integrals of Piatetski-Shapiro and Rallis.

We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents:

where d_w is the walk dimension, d_f is the fractal dimension, d_s is the spectral dimension, and (tilde{zeta }) is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if d_f and (tilde{zeta } geqslant 0) exist, then d_w and d_s exist, and the aforementioned equalities hold. Moreover, our primary new estimate (d_{w} geqslant d_{f} + tilde{zeta }) is established for all (tilde{zeta } in mathbb{R}).

For the uniform infinite planar triangulation (UIPT), this yields the consequence d_w=4 using d_f=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and (tilde{zeta }=0) (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020)). The conclusion d_w=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that d_w=d_f for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since d_f>2.

For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space (mathcal{M}_{g}) of all genus g surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the short window limit, we recover GOE statistics for the variance. The proof makes essential use of Mirzakhani’s integration formula.