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

Abstract

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.