39 citations · 66 across the 23 of their papers we have counts for
24 papers
Spherical harmonics with maximal Lp norm growth
Xiaolong Han
In this paper, we show that for every p>2, there is an orthonormal basis of spherical harmonics on Sn with n>=2 which achieves the maximal Lp norm growth. More precisely, one basis…
Growth of Simple Closed Geodesics in Finite-Volume Hyperbolic Manifolds
Xiaolong Hans Han, Bohan Yang
Let be a finite-volume hyperbolic -manifold with . We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than…
Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds
Xiaolong Hans Han, Ruojing Jiang
We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold contains a sequenc…
Counting surface subgroups in cusped hyperbolic 3-manifolds
Xiaolong Hans Han, Zhenghao Rao, Jia Wan
Let be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of (up to conjugacy and commensurabili…
Large Steklov eigenvalues on hyperbolic surfaces
Xiaolong Hans Han, Yuxin He, Han Hong
In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue tends to infinity…
Spherical harmonics and point configurations on the sphere
Xiaolong Han
We construct spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and…
Fractal uncertainty principle for discrete Cantor sets with random alphabets
Suresh Eswarathasan, Xiaolong Han
In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M di…
Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic -Manifolds
Xiaolong Hans Han
We bound the -norm of an harmonic -form in an orientable cusped hyperbolic -manifold by its topological complexity, measured by the Thurston norm, up to a cons…
On Chemical Distance and Local Uniqueness of a Sufficiently Supercritical Finitary Random Interlacement
Zhenhao Cai, Xiao Han, Jiayan Ye +1
In this paper, we study geometric properties of the unique infinite cluster in a sufficiently supercritical Finitary Random Interlacements in $\mathbb{Z}^d…
From nodal points to non-equidistribution at the Planck scale
Xiaolong Han
In this note, we make an observation that Laplacian eigenfunctions fail equidistribution at the Planck scale. Furthermore, equidistribution at the same scale also fails around the…
Small scale quantum ergodicity in cat maps. II. Quasimodes that are not equidistributed at the logarithmical scales
Xiaolong Han
In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps'"). In Part II of the series, we construct quasimodes that are…
On Eigenvalues of Geometrically Finite Hyperbolic Manifolds with Infinite Volume
Xiaolong Hans Han
Let be an oriented geometrically finite hyperbolic manifold of infinite volume with dimension at least . For all , we provide a lower bound on the th eigenvalue…
Riemann moduli spaces are quantum ergodic
Dean Baskin, Jesse Gell-Redman, Xiaolong Han
In this note we show that the Riemann moduli spaces equipped with the Weil--Petersson metric are quantum ergodic for . We also provide other examples of sin…
Small scale quantum ergodicity in cat maps. I
Xiaolong Han
In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at vari…
Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds. I
Xiaolong Han
In this paper, we investigate the Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds . Via the spectral measure of the Laplacia…
Nodal lengths of eigenfunctions in the disc
Xiaolong Han, Michael Murray, Chuong Tran
In this paper, we derive the sharp lower and upper bounds of nodal lengths of Laplacian eigenfunctions in the disc. Furthermore, we observe a geometric property of the eigenfunctio…
Distribution of the nodal sets of eigenfunctions on analytic manifolds
Xiaolong Han
The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distributi…
Equidistribution of random waves on small balls
Xiaolong Han, Melissa Tacy
In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small s…
Small scale equidistribution of random eigenbases
Xiaolong Han
We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group of isometries acts transi…
Small scale quantum ergodicity in negatively curved manifolds
Xiaolong Han
The main theorem has been slightly generalized to include a larger class of symbols.
Spherical harmonics with maximal Lp (2<p<=6) norm growth
Xiaolong Han
In this paper, we show that there exists a positive density subsequence of orthonormal spherical harmonics which achieves the maximal Lp norm growth for 2<p<=6, therefore giving an…
Sharp norm estimates of layer potentials and operators at high frequency
Jeffrey Galkowski, Xiaolong Han, Melissa Tacy
In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency . For single layer potentia…
Completeness of boundary traces of eigenfunctions
Xiaolong Han, Andrew Hassell, Hamid Hezari +1
In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner, exp…
Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group
Xiaolong Han
In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. On one hand, we apply the concentration compactness principle to prove the ex…