Perhaps I am overlooking something, but a positive answer to your question looks rather straightforward to me, provided the Laplace-Beltrami operator on the manifold is defined suitably -- i.e., weakly.

Consider the closed quadratic form $a(u):=\int_M |\nabla u|^2 \ d\sigma$ for $u\in H^1(M)$. The associated operator is $\Delta$ and it generates an analytic semigroup by the general theory of linear semigroups. Voilà.

(Regularity in space is a more subtle issue, due to the fact that analytic semigroups enjoy very good smoothness with respect to the domains of their generators' powers, hence typically in terms of Sobolev spaces; but on a singular manifold Sobolev imbeddings may be hard to prove.)

Perhaps I am overlooking something, but a positive answer to your question looks rather straightforward to me, provided the Laplace-Beltrami operator on the manifold is defined suitably -- i.e., weakly.

Consider the closed quadratic form $a(u):=\int_M |\nabla u|^2 \ d\sigma$ for $u\in H^1(M)$. The associated operator is $\Delta$ and it generates an analytic semigroup by the general theory of linear semigroups. Voilà.

(Regularity in space - and in particular the possibility to go back from a weak formulation of the parabolic equation to the strong one -- is a more subtle issue, due to the fact that analytic semigroups enjoy very good smoothness with respect to the domains of their generators' powers, hence typically in terms of Sobolev spaces; but on a singular manifold Sobolev imbeddings may be hard to prove.)