Some Remarks on the Heat Flow for Functions and Forms

Abstract

This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation dP t f = P t df is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown that well-known properties which hold on complete Riemannian manifolds fail if the manifold is only BM-complete. In general, even if M is flat and f smooth of compact support, dP t f ∞ cannot be estimated on compact time intervals in terms of f or df .

Heat Equation Derivative Formulas for Vector Bundles

“…On a complete manifold, by the spectral theorem, one has dP t a=P t da, and dual to this, d *P t a=P t d *a, see Section B.2 in Appendix B below. If we drop completeness then these equations are no longer true, even if M is BM-complete, see [59]. But we will show that there always exist Bismut type formulas for dP t a and d *P t a, not involving derivatives of a, independently whether a is smooth or not.…”

Heat flow regularity, Bismut–Elworthy–Li’s derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature

“…(2.4) If one drops geodesic completeness of M , such a commutation relation becomes subtle (cf. [62] for a negative and [32] for a positive result in this direction).…”

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations: part II

“…For a complete Riemannian manifold, a sufficient condition for this to hold is that the Ricci curvature is bounded from below, see e.g. [22] and [19,Eq 1.4]. We are not able to give such a simple formulation for the sub-Laplacian, however, we are able to prove that it holds in many cases, including fiber bundles with compact fibers and totally geodesic fibers.…”

“…However, even if we know that P t 1 = 1 and that the manifold is flat, condition (A) still may not hold if g is an incomplete metric. See [19] for a counter-example.…”