Related papers: Sharp resolvent estimates on non-positively curved…

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We…

A proof for the lower bound is provided for the smallest eigenvalue of finite element equations with arbitrary conforming simplicial meshes. The bound has a similar form as the one by Graham and McLean [SIAM J. Numer. Anal., 44 (2006), pp.…

For a Riemannian closed spin manifold and under some topological assumption (non-zero $\hat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the…

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(\lambda)= \kappa_0\lambda^d…

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base $\Omega\subset \mathbb{R}^d$ and a time-dependent separation $\Lambda$. Under certain mild regularity…

We examine semiclassical magnetic Schr\"{o}dinger operators with complex electric potentials. Under suitable conditions on the magnetic and electric potentials, we prove a resolvent estimate for spectral parameters in an unbounded parabolic…

We first prove semiclassical resolvent estimates for the Schr{\"o}dinger operator in R d , d $\ge$ 3, with real-valued potentials which are H{\"o}lder with respect to the radial variable. Then we extend these resolvent estimates to exterior…

The Stokes resolvent problem $\lambda u - \Delta u + \nabla \phi = f$ with $\mathrm{div}(u) = 0$ subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that…

Labourie raised the question of determining the possible asymptotics for the growth rate of compact $k$-surfaces, counted according to energy, in negatively curved $3$-manifolds, indicating the possibility of a theory of thermodynamical…

We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation…

We study the analytic torsion of odd-dimensional hyperbolic orbifolds $\Gamma \backslash \mathbb{H}^{2n+1}$, depending on a representation of $\Gamma$. Our main goal is to understand the asymptotic behavior of the analytic torsion with…

We investigate the asymptotic spectral distribution of the twisted Laplacian associated with a real harmonic 1-form on a compact hyperbolic surface. In particular, we establish a sublinear lower bound on the number of eigenvalues in a…

We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a…

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…

Let $M$ be a complex manifold and $L$ be a line bundle over $M$ with a Hermitian metric $h$ whose Chern form is a K\"ahler form $\omega$. Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$. When $X$ satisfies the Bohr-Sommerfeld…

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients…

In work with P. Chru\'sciel, L. Nguyen and T.-T. Paetz [8], a positive mass theorem was obtained for asymptotically locally hyperbolic manifolds with boundary, having a toroidal end. The proof made use of properties of marginally outer…

We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr-de Sitter black holes. We recover the optimal…

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the…

Let $\Lambda^{\mathbb{R}}$ denote the linear space over $\mathbb{R}$ spanned by $z^{k}$, $k \in \mathbb{Z}$. Define the real inner product (with varying exponential weights) $<\boldsymbol{\cdot},\boldsymbol{\cdot} >_{\mathscr{L}} \colon…