Abstract
Let M be a complete Riemannian manifold possibly with a boundary ∂M. For any C1-vector field Z, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L:= Δ+Z, pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of ∂M if it exists. These characterizations extend and strengthen the recent results derived by Naber for the uniform norm ∥RicZ∥∞ on manifolds without boundaries. A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author, such that the proofs are significantly simplified.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11771326 and 11431014). The authors thank the referees for the helpful comments
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Wang, F., Wu, B. Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds. Sci. China Math. 61, 1407–1420 (2018). https://doi.org/10.1007/s11425-017-9296-8
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DOI: https://doi.org/10.1007/s11425-017-9296-8