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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?

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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

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  • $\begingroup$ Thanks a lot for a great answer! Do you have a reference for the relationship between spectral gaps and heat kernels? I do not see this disucssed in the Wikipedia page. $\endgroup$
    – Sven Mortenson
    Commented Apr 24, 2022 at 13:17

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