First nonzero eigenvalue of the Laplacian on the submanifold

Question

Consider a compact, connected $n$ dimensional Riemmanian manifold $\mathcal{N}$ and its $m$ dimensional closed submanifold $\mathcal{M}$ (with the metric coming from from the one defined on $\mathcal{N}$).

Question: Is there any way to give a lower bond for the first nonzero eigenvalue of Laplace- Beltrami operator ($\Delta f = - div(grad f)$) defined on $\mathcal{M}$? In particular, is such lower bound related to the first nonzero eigenvalue of the Laplacian on $\mathcal{N}$?

I am especially interested in in the case when $\mathcal{M}$ is of "small" codimension and is given by the intersection of level sets of some smooth functions defined on $\mathcal{N}$. The original motivation for this question comes from the problem of "incheritance" of measure concentration by $\mathcal{M}$, when it is known that it occours for $\mathcal{N}$.

Answer 1

In the general case (especially for high codimension) there will not be such a relation: Every compact Riemannian manifold can be isometrically embedded into the Euclidean space $R^N$. As the image of the submanifold is compact, you can change the metric "near infinity" such that it defines a metric on the sphere $S^N.$

But in special cases, there might be a relation, see for example Yau's conjecture: The first non-zero eigenvalue of a compact embedded minimal hypersurface in (the round) $S^n$ is n. You should take a look into the paper of S. Montiel & A. Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal area, Invent Math.