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A Myers theorem via m-Bakry-Émery curvature. (English) Zbl 1314.53072
The article extends the Myers Theorem [J. Cheeger and 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); J. Cheeger et al., J. Differ. Geom. 17, 15–53 (1982; Zbl 0493.53035)] to complete manifolds whose \(m\)-Bakry-Émery curvature satisfies \[ \mathrm{Ric}_{f,m}(x)\geq -(m-1)\frac{K_{0}}{(1+r(x))^{2}},\tag{1} \] {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 \[ \mathrm{diam}_{M}\;<\;2(e^{\frac{2\pi}{\bar{K}}}-1) \] 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 \[ \triangle_{f}=\triangle_{g}-\nabla f.\nabla \] satisfies the identity \[ \int_{M}\nabla u.\nabla vd\mu=-\int_{M}u.\triangle_{f}vd\mu, \] for any \(u,v\in C^{\infty}_{0}(M)\).
{Definition}: The \(m\)-Bakry-Émery curvature is
\[ \mathrm{Ric}_{f,m}=\mathrm{Ric}_{g}+\mathrm{Hess}(f)-\frac{df\otimes df}{m-n}. \]
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.

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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