A Riemannian Off-Diagonal Heat Kernel Bound for Uniformly Elliptic Operators

Riemannian Manifolds With Uniformly Bounded Eigenfunctions

The standard eigenfunctions $ϕ_λ = e^{i < λ, x >}$ on flat tori $\R^n / L$ have $L^{\infty}$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $L^{\infty}$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with completely integrable geodesic flows.

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

We prove two explicit bounds for the multiplicities of Steklov eigenvalues $σ_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues $σ_k$ are uniformly bounded in $k$.

Discretization of Riemannian manifolds applied to the Hodge Laplacian

An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of balls of sufficiently small radius). We exhibit then a lower bound for the first positive eigenvalue of the combinatorial Laplacian and deduce a lower bound for the first positive eigenvalue of the Hodge Laplacian.

A Hardy inequality on Riemannian manifolds and classification of discrete Dirichlet spectra

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion under which the Laplace- Beltrami operator with Dirichlet boundary conditions $Δ^D$ has purely discrete spectrum on $Ω$. We proceed to classify all non-compact $Ω$ with discrete spectrum up to a boundary regularity condition and show that these include for example polygons with ide…

黎曼流形上拉普拉斯算子的图离散化： Ricci曲率有界条件下

Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature

本文在Ricci曲率有界条件下，通过图近似方法证明了流形拉普拉斯算子特征值的均匀收敛性。

Isoperimetric control of the Steklov spectrum

Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of the genus. We also obtain a relationship bet…