Compactness in Weighted Manifolds and Applications

Compactness theorems for the Bakry-Emery Ricci tensor on semi-Riemannian manifolds

“…where (∇f ) ⊥ is a normal projection of ∇f on M . We will use the following technical lemma obtained in [7] (see also [5]).…”

“…In the Riemannian context, the classical Myers compactness theorem [15] was extended for weighted manifolds in many ways. See for instance, [2], [5], [11], [12], [13], [14], [16], [17], [21], [24]. Exploring a Riccati inequality obtained from the Bochner formula we get the following improved version of weighted Myers theorem in the spirit of the Sprouse theorem in [20].…”

“…In this sense, the works of Galloway in [8] and Galloway and Frankel in [7] are very successful. Recently, Cavalcante, Oliveira and the author [5] generalized the closure theorems of [7] and [8] in the weighted case. Here, using Theorem 1.1 we are able to improve Theorem 4.3 of [5] as follows: Theorem 1.3.…”

“…Recently, Cavalcante, Oliveira and the author [5] generalized the closure theorems of [7] and [8] in the weighted case. Here, using Theorem 1.1 we are able to improve Theorem 4.3 of [5] as follows: Theorem 1.3. Let M n be a spatial hypersurface inM n+1 f and assume that M is complete in the induced metric.…”

Myers’ type theorem with the Bakry–Émery Ricci tensor

“…Remark 1.2. There have been other Myers' type theorems involving the Bakry-Émery Ricci tensor under different conditions; see Bakry and Qian [3], Cavalcante, Oliveira and Santos [6], Li [12], Limoncu [14], Lott [15], Mastrolia, Rimoldi and Veronelli [16], Morgan [17], Soylu [21], Tadano [22], Wei and Wylie [25] and Zhang [26] for details.…”