Half-Space Theorems for Minimal Surfaces with Bounded Curvature

Abstract

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of R 3 . We also show that any minimal hypersurface immersed with bounded curvature in MˆR`equals some Mˆtsu provided M is a complete, recurrent n-dimensional Riemannian manifold with Ric M ě 0 and whose sectional curvatures are bounded from above. For H-surfaces we prove that a stochastically complete surface M can not be in the mean convex side of a H-surface N embedded in R 3 with bounded curvature if sup |H M | ă H, or distpM, N q " 0 when sup |H M | " H. Finally, a maximum principle at infinity is shown assuming M has non-empty boundary.

A martingale approach to minimal surfaces

“…Let ρ = x 2 1 + x 2 2 + x 2 3 . Then a straightforward computation shows (see [14,Section 5.2.2]) that 1 2 ρ 2 = 2. It follows from the defining property of the martingale problem that E ρ 2 t∧e = ρ 2 0 + 2(t ∧ e) for all t ∈ [0, ∞).…”

Structure theorems for embedded disks with mean curvature bounded in $L^p$

“…If Σ n has another accumulation point which is not in Σ, then the same argument shows that there exists another complete connected properly embedded minimal surface, Σ ′ , which is contained in the accumulation set of Σ n and it is disjoint from Σ. The results in [1,11,21] imply that they must be parallel planes.…”

On Bernstein–Heinz–Chern–Flanders inequalities

“…We observe here that the proof of Theorem 1.5 of [6] can be straight forward adapted to give a version Theorem 3.1 of Barbosa-Kenmotsu-Oshikiri [3] for complete Riemannian manifolds M nonnegative Ricci curvature. We have the following theorem.…”

“…We reproduce here the proof of Theorem 1.5 of [6] with proper modifications that yields the proof of Theorem 1.5. Let F be transversely oriented codimension-one C 2 -foliation of a complete Riemannian manifold M with bounded geometry and nonnegative Ricci curvature Ric M ≥ 0.…”