Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a **spectral gap** for such a $\Delta$, that is, finding an interval $(0,c)$ which does not contain any eigenvalues of $\Delta$. Why are people interested in producing such spectral gaps? What interesting information do they give about the manifold? Are there interesting applications in the `real work', i.e. in physics?



Why are we interested in spectral gaps for Laplacian operators
A spectral gap gives information on geometry of the manifold via Cheeger's inequality, <https://en.wikipedia.org/wiki/Cheeger_constant> See also Buser's inequality discussed there. More directly, a spectral gap for the Laplacian yields exponential decay for the heat kernel and determines the asymptotic rate of decay. The heat kernel represents transition probabilities for Brownian motion on the manifold, so a spectral gap means the motion mixes rapidly (in the finite volume case) and dissipates quickly (in the infinite volume case).


<https://en.wikipedia.org/wiki/Heat_kernel>


