Geometric and Functional Analysis最新文献

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.

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let (H: L^{2}(mathbb{R})to L^{2}(mathbb{R})) have the form

where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, ({1}_{(-infty ,rho ^{2}]}(H)), has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.

We consider the space of ordered pairs of distinct ({mathbb{C}{mathrm{P}}}^{1})-structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal.

In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasi-conformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties ((operatorname{SL}_{2}mathbb{C})-opers) is a non-empty discrete set, which is closely related to the mapping.

We find the generating function for the contributions of n-cylinder square-tiled surfaces to the Masur–Veech volume of (mathcal{H}(2g-2)). It is a bivariate generalization of the generating function for the total volumes obtained by Sauvaget via intersection theory. Our approach is, however, purely combinatorial. It relies on the study of counting functions for certain families of metric ribbon graphs. Their top-degree terms are polynomials, whose (normalized) coefficients are cardinalities of certain families of metric plane trees. These polynomials are analogues of Kontsevich polynomials that appear as part of his proof of Witten’s conjecture.

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.