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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.

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