English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let M2 be a complete simply connected Riemannian surface with curvature KM2 ≤ −κ ≤ 0, for some constant κ (i.e., M2 is a Hadamard surface). We consider a compact CMC (constant mean curvature) H surface Σ embedded in M2×R, with Γ = ∂Σ ⊂ Q = M2 × {0}. We obtain a relation between the area of a part of Σ above Q, the height above Q of this part, and the volume it bounds. As a corollary of this, when ∂Σ = ∅ we obtain: 2HA ≥ κV + 4πh. Our estimate is inspired by the paper [5]. Moreover we prove that if Σ is a compact H-surface embedded in H2×R whose boundary, Γ = ∂Σ, is a convex planar curve contained in Q = H2 × {0} and the height of the surface Σ is less than or equal the height of the hemisphere of the rotational sphere with constant mean curvature H, then Σ stays in a half-space determined by Q and is transverse to Q along the boundary and inherits the same symmetries of its boundary. We conclude using these two results that if Σ is a compact H-surface embedded in H2×R with convex planar boundary and the height of the surface Σ is less than or equal to the height of the hemisphere of the rotational sphere with constant mean curvature H, then Σ is a graph. The equality holds if and only if Σ is the rotational H-hemisphere.

A relation between height, area, and volume for compact constant mean curvature surfaces in M2 x R