A comparison theorem for an elliptic operator
Let \(\varrho(x)\) be the distance from a fixed reference point in a manifold. Using the Bakry-Emery technique, the author proves a comparison theorem similar to the upper bound for \(\Delta\varrho\) on a Riemannian manifold with lower bounded Ricci curvature. More precisely, consider a second-order elliptic differential operator \(L\) on a smooth manifold \(M\). The diffusion part of \(L\) gives rise to a Riemannian structure on \(M\), a corresponding carré du champ-operator \(\Gamma(f, g)={1\over 2}(L(fg)- fLg- gLf)\) and the Bakry-Emery curvature \(\Gamma_2(f,g)= {1\over 2}(L\Gamma(f,g)- \Gamma(Lf,g)- \Gamma(f, Lg))\). The author assumes that \(M\) is complete and that \(L\) satisfies the so-called curvature-dimension inequality \(\Gamma_2(f,f)\geq {1\over n}(Lf)^2- K\Gamma(f,f)\). In the case of the Laplace-Beltrami operator on an \(n\)-dimensional manifold, this is equivalent to a lower bound \(-K\) for the Ricci curvature. Under these assumptions, the author proves that on the complement of the cut-locus of the reference point the following comparison result holds: \[ L\varrho^2\leq n\Biggl\{1+ \sqrt{1+ {4K\varrho^2\over n}}\Biggr\}. \]
scientific article; zbMATH DE number 1160074