Riemann manifold with unbounded Laplacian

Question

How can one characterize a Riemann manifold the Laplacian of which is unbounded? (Equivalently, what are those manifolds on which the Laplacian is bounded? I am interested in working with its exponential.)

Answer 1

The Laplacian should essentially always be unbounded in the sense that its eigenvalues are unbounded. Loosely, one can think of eigenvalues of the Laplacian as frequencies at which waves can propagate in the manifold, and it's not surprising that waves can propagate at arbitrarily high frequencies.

More precisely, on a compact Riemannian $d$-manifold, Weyl's law implies that the $n^{th}$ eigenvalue of the Laplacian grows in absolute value something like $O(n^{2/d})$. For example, on $S^1$ the $n^{th}$ eigenvalue has absolute value $n^2$.

However, one can make sense of the exponential $e^{t \Delta}$ when $t$ is positive (here by $\Delta$ I mean the negative-semidefinite Laplacian). See heat kernel for some details.