Related papers: Geodesic restrictions for the Casimir operator

We investigate the asymptotic behavior of high-codimensional area-minimizing locally rectifiable currents in hyperbolic space, addressing a problem posed by F.H. Lin and establishing ``boundary regularity at infinity" results for such…

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}:…

We extend the Feynman-Kac formula for Schr\"odinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be…

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula…

Let $R$ be a compact Riemann surface and $\Gamma$ be a Jordan curve separating $R$ into connected components $\Sigma_1$ and $\Sigma_2$. We consider Calder\'on-Zygmund type operators $T(\Sigma_1,\Sigma_k)$ taking the space of $L^2$…

In this paper, we establish the essential criteria for the hyponormality and quasinormality of the unbounded Toeplitz operator $T_{\varphi}$ with non-harmonic symbol, acting on the Fock-Sobolev space $F^{2, m}(\mathbb{C})$. The study shows…

We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in all dimensions $n\ge3$ on general manifolds. On manifolds of constant sectional curvature, we introduce a…

We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…

We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and…

A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity…

In this paper, we prove the uniform estimates for the resolvent $(\Delta - \alpha)^{-1}$ as a map from $L^q$ to $L^{q'}$ on real hyperbolic space $\mathbb{H}^n$ where $\alpha \in \mathbb{C}\setminus [(n - 1)^2/4, \infty)$ and $2n/(n + 2)…

For smooth bounded pseudoconvex domains in $mathbb{C}^{2}$, we provide geometric conditions on (the points of infinite type in) the boundary which imply compactness of the $\bar{\partial}$-Neumann operator. It is noteworthy that the proof…

Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.

For each k > 0 we find an explicit function f_k such that the topology of S inside the ball B(p,r) is `bounded' by f_k(r) for every complete Riemannian surface (compact or noncompact) with K\geq -k^2, every point p on the surface, and every…

In the setting of spaces of homogeneous type, we give a direct proof of the local Tb theorem for singular integral operators. Motivated by questions of S. Hofmann, we extend it to the case when the integrability conditions are lower than 2,…

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of…

If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the…

We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the…

We give a lower bound for the bottom of the $L^2$ differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent…

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is shown that this operator has a…