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Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds. (English) Zbl 1400.58011

Summary: Let \(M\) be a complete Riemannian manifold possibly with a boundary \(\partial M\). For any \(C^1\)-vector field \(Z\), by using gradient/functional inequalities of the (reflecting) diffusion process generated by \(L:= \Delta+Z\), pointwise characterizations are presented for the Bakry-Emery curvature of \(L\) and the second fundamental form of \(\partial M\) if it exists. These characterizations extend and strengthen the recent results derived by Naber for the uniform norm \(\|\mathrm{Ric}_Z\|_\infty\) 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.

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J65 Diffusion processes and stochastic analysis on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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[1] Arnaudon, M; Thalmaier, A; Wang, F Y, Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds, Stochastic Process Appl, 119, 3653-3670, (2009) · Zbl 1178.58013 · doi:10.1016/j.spa.2009.07.001
[2] Besse A L. Einstein Manifolds. Berlin: Springer, 1987 · Zbl 0613.53001 · doi:10.1007/978-3-540-74311-8
[3] Capitaine, B; Hsu, E P; Ledoux, M, Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces, Electron Comm Probab, 2, 71-81, (1997) · Zbl 0890.60045 · doi:10.1214/ECP.v2-986
[4] Chen, X; Wu, B, Functional inequality on path space over a non-compact Riemannian manifold, J Funct Anal, 266, 6753-6779, (2014) · Zbl 1307.58014 · doi:10.1016/j.jfa.2014.03.017
[5] Driver, B, A cameron-martin type quasi-invariant theorem for Brownian motion on a compact Riemannian manifold, J Funct Anal, 110, 272-376, (1992) · Zbl 0765.60064 · doi:10.1016/0022-1236(92)90035-H
[6] Fang, S Z, Inégalité du type de Poincaré sur l’espace des chemins riemanniens, C R Math Acad Sci Paris, 318, 257-260, (1994) · Zbl 0805.60056
[7] Haslhofer R, Naber A. Ricci curvature and Bochner formulas for martingales. ArXiv:1608.04371, 2016 · Zbl 1393.60042
[8] Hsu, E P, Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds, Comm Math Phys, 189, 9-16, (1997) · Zbl 0892.58083 · doi:10.1007/s002200050188
[9] Hsu, E P, Multiplicative functional for the heat equation on manifolds with boundary, Michigan Math J, 50, 351-367, (2002) · Zbl 1037.58024 · doi:10.1307/mmj/1028575738
[10] Naber A. Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces. ArXiv:1306.6512v4, 2013
[11] Wang, F Y, Weak Poincaré inequalities on path spaces, Int Math Res Not IMRN, 2004, 90-108, (2004) · Zbl 1083.58034 · doi:10.1155/S1073792804130882
[12] Wang, F Y, Analysis on path spaces over Riemannian manifolds with boundary, Commun Math Sci, 9, 1203-1212, (2011) · Zbl 1282.60079 · doi:10.4310/CMS.2011.v9.n4.a14
[13] Wang F Y. Analysis for Diffusion Processes on Riemannian Manifolds. Singapore: World Scientific, 2014 · Zbl 1296.58001
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