Maximum principles on Riemannian manifolds and applications

Rigid properties of quasi-Einstein metrics

“…Using the maximum principle [15,16,17], we can get the gradient estimate of the potential function f , of independent interest. …”

Submanifolds with nonpositive extrinsic curvature

“…For the proof of this, as well as of the other results related to the Omori-Yau maximum principle in this subsection, we refer to [14]. …”

“…The Chern and Kuiper result gave rise to a long series of works, among others, by O'Neill [12], Stiel [19], Moore [9], Jorge-Koutroufiotis [6], Pigola-Rigoli-Setti [14] and, finally, by Alías-Bessa-Montenegro [3] who obtained Theorem 1 on cylindrically bounded submanifolds.…”

“…The maximum principles used throughout this paper, as well as their related results, namely, Theorems 8, 9 and 10, are due to Pigola-Rigoli-Setti [14]. On the other hand, it was shown in [1] Regarding complete hypersurfaces of nonpositive Ricci curvature, Leung [8] pioneered their study by proving Theorem 3 in the case b = 0 and conjecturing that the assumption on the sectional curvature could be dropped.…”

“…On the other hand, a key ingredient to handle the noncompact case is a maximum principle due to Omori [11] and generalized by Pigola-Rigoli-Setti [14]. Using this principle, Alías-Bessa-Dajczer [2] obtained an estimate for the mean curvature of an isometric immersion f : M m → N n+l = P n × R l , under some assumptions on the manifold P n , whose projection onto the first factor is bounded, the socalled cylindrically bounded submanifolds.…”