Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$. 


In pages 10-11 of [this paper](http://www.ma.utexas.edu/mp_arc/c/09/09-101.pdf), the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the problem
$$\Delta\_{\overline g}(u) = 0\quad\text{on $M\times (0,\infty)$}$$
$$u(x,0) = u\_0(x) \quad \text{on $M \times \{0\}$}.$$


Here $\overline g$ is the metric on $\mathcal M$. Then they define the Dirchlet to Neumann map $\mathcal L(u\_0) = \partial\_{\nu}(u)|\_{x=0}$ where $x$ represents the coordinates on $(0,\infty)$. They conclude in formula (3.31) that $$\mathcal L = (-\Delta\_g)^{\frac 12}$$
is the fractional Laplacian, where $\Delta\_g$ is the Laplace-Beltrami.


So then this is a definition of the fractional Laplacian on compact hypersurfaces (and Riemannian manifolds) if I understand it correctly. However, I have not seen this definition anywhere else and can find no other reference to it, and there appears to be some disagreement that this is correct (if I understand correctly) in the comments of this post. Can someone perhaps clarify this? Thanks.



Fractional Laplacian on compact hypersurface/manifold via harmonic extension?
The relation between the fractional Laplacian and the Dirichlet-to-Neumann map was considered by [Caffarelli and Silvestre](http://www.ams.org/mathscinet-getitem?mr=2354493) for the $\mathbb{R}^n$ case. This notion is crucial in the study of [nonlocal minimal surfaces](http://www.ams.org/mathscinet-getitem?mr=2675483). 


The Caffarelli-Silvestre construction was generalised by [Chang and Gonzalez](http://www.ams.org/mathscinet-getitem?mr=2737789), which is largely the standard reference now in that direction. (One of the neat things in the Chang and Gonzalez paper is the generalisation to other important operators in conformal geometry, such as the Paneitz and GJMS operators.) This has lead to the study of problems such as the [fractional Yamabe problem](http://arxiv.org/abs/1012.0579).


For a reference request, the above should give you a bit of stuff to start with. 


You can find other references on the [nonlocal Wiki](https://www.ma.utexas.edu/mediawiki/index.php/Starting_page). 


