A Myers theorem via \textit{m}-Bakry-Émery curvature (Q2452692)

The article extends the Myers Theorem [\textit{J. Cheeger} and \textit{D. G. Ebin}, Comparison theorems in Riemannian geometry. Amsterdam-Oxford: North-Holland Publishing Company; New York: American Elsevier Publishing Company, Inc. (1975; Zbl 0309.53035); \textit{J. Cheeger} et al., J. Differ. Geom. 17, 15--53 (1982; Zbl 0493.53035)] to complete manifolds whose \(m\)-Bakry-Émery curvature satisfies \[ \mathrm{Ric}_{f,m}(x)\geq -(m-1)\frac{K_{0}}{(1+r(x))^{2}},\tag{1} \] \noindent {Main Theorem}: Assume the \(m\)-Bakry-Émery curvature satisfies condition (1) for some constant \(K_{0}<-\frac{1}{4}\). Then the manifold \(M\) is compact and its diameter satisfies \[ \mathrm{diam}_{M}\;<\;2(e^{\frac{2\pi}{\bar{K}}}-1) \] \noindent where \(\bar{K}=\sqrt{-K_{0}}\). Let \((M,g)\) be an \(n\)-dimensional complete manifold and \(f:M\rightarrow\mathbb{R}\) a smooth function named potential. By considering on \(M\) the weighted measure \(d\mu=e^{-f}dV\), where \(dV\) is the Riemannian-Lebesgue measure induced on \(M\) by \(g\), the weighted Laplacian \[ \triangle_{f}=\triangle_{g}-\nabla f.\nabla \] \noindent satisfies the identity \[ \int_{M}\nabla u.\nabla vd\mu=-\int_{M}u.\triangle_{f}vd\mu, \] \noindent for any \(u,v\in C^{\infty}_{0}(M)\). \noindent {Definition}: The \(m\)-Bakry-Émery curvature is \[ \mathrm{Ric}_{f,m}=\mathrm{Ric}_{g}+\mathrm{Hess}(f)-\frac{df\otimes df}{m-n}. \] \noindent The author stresses the difficulty to apply the index Lemma, as applied in [Cheger et al., loc. cit.], to the \(m\)-Bakry-Émery curvature case. He goes around using the weighted Laplacian comparision theorem and the excess function.