The p-parabolicity under a decay assumption on the Ricci curvature

2023/10/18 by Priebe, Lucas S., Soares, Rodrigo B.
#35J93 #58J05 #58J32 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.12257

Abstract

We prove that, given α>0, if M is a complete Riemannian manifold which Ricci curvature satisfies.\operatorname*Ric\nolimits_x(v)≥αsech² (r(x))) or \operatorname*Ric\nolimits_x(v)≥-\frach_α (r(x))r(x)², where h_α(r) = \fracα(α+1)r(x)^αr(x)^α-1, for all x∈ M\backslash B_R(o) and for all v∈ T_xM, \Vert v\Vert =1, where o is a fixed point of M, r(x)=d(o,x), d the Riemannian distance in M and B_R(o) the geodesic ball of M centered at o with radius R>0, then M is p-parabolic for any p>1, if satisfies the first inequality, and M is p-parabolic, for any p≥(α+1)(n-1)+1, if satisfies the second inequality.

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