Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature

Abstract

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-Émery-Ricci curvature if the space of weighted L 2 harmonic one-forms is non-trivial then the weighted volume of the manifold is finite and universal cover of the manifold splits isometrically as the product of the real line with an hypersurface.

Rigidity of complete spacelike hypersurfaces with constant weighted mean curvature

“…On the other hand, in the case that n is complete noncompact with Ric f ≥ 0, Theorem 1.1 of Vieira (2013) asserts that a sufficient condition for n to have finite f -volume is that the space of L 2 f harmonic one-forms on n be nontrivial.…”

Harmonic Aspects in an $\eta$-Ricci Soliton

“…The condition λ|γ 0 | 2 + µ(γ 0 (ξ)) 2 ≤ 0 is equivalent to S ♯ f (γ 0 , γ 0 ) ≥ 0. From (29) and Lemma 3.2 from [25]:…”

“…Following the same steps as in [38], we obtain the conclusion. Remark 3.4. i) Under the hypothesis of Theorem 3.4, in particular, we deduce that γ 0 is ∇-parallel and of constant length.…”

$L^2_f$ harmonic 1-forms on smooth metric measure spaces with positive $λ_1(Δ_f)$

“…As a corollary, Vieira obtained (Corollary 1.2, [Vie13]) with the same curvature assumption, if the first eigenvalue of the f -Laplacian is positive, then the space of L 2 f harmonic 1-forms is trivial. Inspired by Li-Wang, Vieira, Dung and Dung-Sung's work, in this paper, we extend Vieira's Theorem 1.4 by relaxing the curvature condition to be Ric m,n ≥ −aλ 1 (∆ f ) without positivity restriction on λ 1 (∆ f ), and generalize Dung-Sung's Theorem 1.2 to complete non-compact smooth metric measure spaces, which can also be considered as an extension of Dung's Theorem 1.3.…”

“…Theorem 1.4 (Theorem 1.1, [Vie13]). Let (M, g, e −f dv) be a complete non-compact smooth metric measure space with non-negative ∞-Bakry-Émery Ricci curvature.…”