On Minimal Surfaces Satisfying the Omori–yau Principle

Abstract

Complete minimal immersions satisfying the Omori-Yau maximum principle are investigated. It is shown that the limit set of a proper immersion into a convex set must be the whole boundary of the convex set. In case of a nonproper and nonplanar immersion we prove that the convex hull of the immersion is a half-space or R 3 .2010 Mathematics subject classification: primary 53C42.

Spectral and stochastic properties of the $f$-Laplacian, solutions of PDEs at infinity and geometric applications

“…We next describe a result which extends a condition on the validity of the full Omori-Yau maximum principle for the operator Δ f proved in [19]. The argument we use is an adaptation of a recent elegant proof of the Omori-Yau maximum principle due to A. Borbély, [5] and [6]. We are grateful to A. Borbély for sending us a copy of [6].…”

“…The argument we use is an adaptation of a recent elegant proof of the Omori-Yau maximum principle due to A. Borbély, [5] and [6]. We are grateful to A. Borbély for sending us a copy of [6].…”

Comparison principle, stochastic completeness and half-space theorems

“…Remark 4 It follows from Borbély's work [11] that this criterion holds without the condition lim sup t→∞ tG(t 1/2 ) G(t) < +∞, see also [9,Thm.9].…”

“…Otherwise, suppose that (10) and (11) in Theorem 2 hold. It follows from the proof of Proposition 1 that…”

“…Suppose that inf U h > 0 and that ϕ(M) ⊂ ρ −1 ((−∞, d)) for some d < +∞. If (10) and (11) in Theorem 2 hold then…”