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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.

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