Related papers: Geodesic restrictions of arithmetic eigenfunctions

We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…

This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in…

Let $\mathcal{M}$ be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm…

Using the amplification technique, we prove that `mass' of the pullback of the Saito-Kurokawa lift of a Hecke eigen form $g\in S_{2k}$ is bounded by $k^{1-\frac{1}{210}+\epsilon}$. This improves the previously known bound $k$ for this…

We prove a quantitative version of the non-uniform hyperbolicity of the Teichm\"uller geodesic flow. Namely, at each point of any Teichm\"uller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the…

On convex co-compact hyperbolic surfaces with Hausdorff dimension of the limit set less than 1/2, we investigate high energy behaviour of Eisenstein Series. Eisenstein Series are non-L^2 eigenfunctions of the hyperbolic Laplacian which…

We prove sub-convex bounds on the fourth moment of Hecke-Laplace eigenforms on $S^3$. As a corollary, we get a bound on the sup-norm on an individual eigenform, which constitutes an improvement over what is achievable through employing the…

The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for…

For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is…

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the…

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we…

This work concerns $L^p$ norms of high energy Laplace eigenfunctions, $(-\Delta_g-\lambda^2)\phi_\lambda=0$, $\|\phi_\lambda\|_{L^2}=1$. In 1988, Sogge gave optimal estimates on the growth of $\|\phi_\lambda\|_{L^p}$ for a general compact…

We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of…

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension…

We consider a local average in the hyperbolic lattice point counting problem for the Picard group $\Gamma$ acting on the three-dimensional hyperbolic space. Compared to the pointwise case, we improve the bounds on the remainder in the…

In this paper we provide a formulation for sweeping processes with arbitrary locally bounded retraction, not necessarily left or right continuous. Moreover we provide a proof of the existence and uniqueness of solutions for this formulation…

We study geodesics for plurisubharmonic functions from the Cegrell class ${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and show that, as in the case of metrics on K\"{a}hler compact menifolds, they linearize an energy…

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the…

Let $X$ be a compact connected orientable hyperbolic surface and let $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the…

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of $\Delta$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic…