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### How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$(z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right).$$ For $z=x+ i y \in \mathbb C$ ...
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### Estimate for the first eigenvalue of the Laplacian

I was studying the paper of S. T. Yau - Seminar on Differential Geometry - and there asks if the first eigenvalue is equal to $n$, if we have a embedded oriented Riemannian manifold and closed ...
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### Obstructions for existence of a Riemannian metric such that a given function is harmonic

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...
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### Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$). Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$. ...
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### Lp norm estimates for the inverse of the Laplacian on a graph

I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in  \sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|...
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### When does the effective concentration of measure does not occour on a Riemmanian manifold?

Introduction Let $\mathcal{M}$ be a compact $m$ - dimensional Riemmanian manifold with normalized measure $\mu$ (derived from the metric). It is know that in this setting we have concentration ...
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### Graph Laplacian to Continuous version

Has there been a study of vector laplacian that is a continuous version of a graph laplacian? Is there a good introduction to the topic?
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### When Spectrum of Laplacian in a non-compact manifold is infinite and discrete?

We know that the spectrum of Laplacian in compact smooth manifolds are discrete and infinite. There is a question about spectrum of Laplacian in non-compact case in mathoverflow, Spectrum of ...