Related papers: Local average in the Hyperbolic sphere problem

We obtain some "universal" estimates for $L_2$-norm of the solution of a parabolic equation via a weighted version of $H^{-1}$-norm of the free term. More precisely, we found the limit upper estimate that can be achieved by transformation…

In this note, we study a lattice point counting problem for spheres in Heisenberg groups, incorporating both the non-isotropic dilation structure and the non-commutative group law. More specifically, we establish an upper bound for the…

We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent…

We work toward the arithmetic quantum unique ergodicity (AQUE) conjecture for sequences of Hecke--Maass forms on hyperbolic $4$-manifolds. We show that limits of such forms can only scar on totally geodesic $3$-submanifolds, and in fact…

We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric…

In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly…

In part II we constructed the lower bound, in the spirit of $\Gamma$- $\liminf$ for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E_\e(v):=\int_\Omega…

The Hubble tension casts a blight on the standard cosmology. As a possible solution to the problem, the local variation of the expansion rate has been proposed where the spatial averaging over a finite domain was introduced in order to…

We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank.

We study a one-dimensional nonlinear hyperbolic-parabolic initial boundary value problem occurring in the theory of thermoelasticity. We prove existence and uniqueness of the local-in-time strong solution. Also, some global-in-time weak…

We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution.In doing so, we put…

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building…

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an…

Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $\Gamma\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms…

We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $\Gamma\curvearrowright G$, whenever $\Gamma$ is a dense…

Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=\Gamma \setminus \HH $ be the corresponding quotient space. For $\gamma,$ a closed geodesic of $M$, let $l(\gamma)$ denote its length. The prime geodesic…

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…

On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies…

In this paper, we establish a lower bound, in terms of the isoperimetric deficit, for the first eigenvalue of the Robin Laplacian with negative boundary parameter on horospherically convex bounded domains in the hyperbolic space. This…

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…