Heat conservation on Riemannian manifolds

Abstract

Let M be an oriented, connected, smooth, Riemannian manifold of dimension n > 1, and let div (κ grad ϕ) = ς ∂ϕ ∂t (1) be the heat equation on M, with initial condition f , where κ > 0 is the conductivity and ς is the heat capacity per unit of volume of M. Let t → T(t) be the semigroup associated to the heat equation (1). If M is compact and d ω is the volume element of M, then M T(t) f d ω = M f d ω, (2) for all t ≥ 0. The case in which the manifold M is open was first considered by L. Gårding, when M is an open subset of R n and f is compactly supported. Gårding's proof was generalized by Gaffney in [5] to the case of a manifold M on which the distance d(o, x) of x ∈ M from a fixed point o ∈ M satisfies the inequality M e −ad(o,x) d ω(x) < ∞, ∀ a > 0.(3)In this paper Gaffney's result will be extended to vector-valued differential forms 1 . More specifically, let the Riemannian manifold M be complete and satisfy (3), let E → M be a Riemannian vector bundle on M, and let T be the semigroup defined by the opposite of the Laplace-Beltrami operator whose domain is dense in the Hilbert space of square-summable q-forms on M with values in E (q = 0, 1, . . . , n).If w is a bounded, smooth, E-valued q-form whose differential and codifferential both vanish, thenfor all t ≥ 0 and all smooth, compactly supported q-forms f with values in E.1 Assuming κ = ς = 1.

Vanishing and conservativeness of harmonic forms of a non-compact CR manifold

“…The following is the key estimate, which was proved for a complete Riemannian manifold by Vesentini [32]. Since M has negligible boundary, (19) and 20 Thus, we have e br/2 ψ ≤ e b √ 2t e br/2 ψ 0 .…”

“…Another issue related to harmonic forms which we study is the conservative principle. The concept of conservative principle for differential forms has been introduced by Vesentini [32], who proved it for complete Riemannian and Kähler manifolds under the volume growth condition (1) (see also [23]). In the present paper, we will study a similar type of conservativeness, which coincides with Vesentini's for a complete Riemannian manifold under certain curvature conditions (see Remark 6 in Section 5), and is rather similar to that of a Dirichlet form [7]: the canonical Dirichlet form on a CR manifold is conservative if there exist…”

“…Let α be a bounded harmonic form. Vesentini [32] proved that a complete Riemannian manifold with volume growth condition (1) satisfies (26) T t ψ, α = ψ, α for every ψ ∈ A q 0 and t > 0,…”

Conservative principle for differential forms

“…Since ∆ M is essentially self-adjoint (e.g. [3], [17], [19]), and since ∆ N ⊂ ∆ M , it suffices to prove that…”

“…While both the stochastic completeness and Hodge theory have been studied with considerable efforts, there has been no notion of conservative principle on A q , until Vesentini's recent work [19]. In that paper, he extended the notion of conservative principle from functions to A q as follows: DEFINITION 1.…”