# Heat-Flow on continuous differential forms and the Feller peroperty

Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $$C^\infty_c(M)$$. This extends to the (de Rahm) Laplace operator on forms. Thus in each case, the heat semi-group is well-defined on $$L^2(M)$$. Let $$P_t$$ denote the semi-group generated by the Laplacian on forms.

Can anything be said about $$\|{P_t f}\|_{L^\infty}$$ as $$t \rightarrow 0$$? By the Weitzenböck formula, we know that $$\Delta = \nabla^*\nabla + R$$, where $$R$$ is the Ricci tensor. If $$R$$ is bounded from below, there are well-known estimates of $$\|P_t f\|_{L^\infty}$$ as $$t \rightarrow 0$$, which I do not expect to hold for general $$R$$. However, since $$M$$ is complete, we know that $$\Delta$$ is self-adjoint, which should restrict the class of admissible $$R$$ somewhat. Is this restriction strong enough that it is possible to gain some control on the $$L^\infty$$ norm for initial data given by a smooth, compactly suppoted $$k$$-form? If not, are there any counterexamples to $$\|{P_t f}\|_{L^\infty} < \infty$$ for small $$t$$, or $$\limsup\limits_{t \rightarrow 0} \|P_t f\|_{L^\infty} \leq \| f \|_{L^\infty}$$?

This seems somewhat related to the question of whether $$(M,g)$$ has the Feller-property, though I'm not sure exactly how. Of course, if $$P_t f$$ converges uniformly to $$f$$ as $$t \rightarrow 0$$, this implies the result, but I don't think this is true in general.

Edit: at the end of his paper "$$L^p$$ contractive projections and the Heat-Semigroup for Differential Forms", Strichartz constructs an example of a (non-connected) Riemannian manifold for which the heat semigroup on 1-forms is unbounded for any $$p \neq 2$$ and $$t > 0$$. So I should probably add connectedness to the list of assumptions.