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2007•International Mathematical ForumOpen access

Gauss curvature estimates for surfaces whose mean curvature is constant or has special properties

Fei-Tsen Liang

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Abstract

Let z = w(x, y) represent an embedded (not necessarily simply-connected), compact nonparametric surface in R3 with mean curvature H, nonpositive Gauss curvature K. Set (−K)max, |H|max to be the global maximum of −K and H and set K0,H0 to be the maximum of −K,H on the boundary. If (logH)xx + (logH)yy ≥ 0, K0> 0, (−K)max ≥ (|H|max), ∂H∂x + ∂H∂y ≥ 0 and | ω | is bounded, then either (|H|max)2 + (−K)max = (H0)2 + K0 or ((|H|max)2 + (−K)max)(H2{x ∈

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Let z = w(x, y) represent an embedded (not necessarily simply-connected), compact nonparametric surface in R3 with mean curvature H, nonpositive Gauss curvature K. Set (−K)max, |H|max to be the global maximum of −K and H and set K0,H0 to be the maximum of −K,H on the boundary. If (logH)xx + (logH)yy ≥ 0, K0> 0, (−K)max ≥ (|H|max), ∂H∂x + ∂H∂y ≥ 0 and | ω | is bounded, then either (|H|max)2 + (−K)max = (H0)2 + K0 or ((|H|max)2 + (−K)max)(H2{x ∈

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Available abstract

Let z = w(x, y) represent an embedded (not necessarily simply-connected), compact nonparametric surface in R3 with mean curvature H, nonpositive Gauss curvature K. Set (−K)max, |H|max to be the global maximum of −K and H and set K0,H0 to be the maximum of −K,H on the boundary. If (logH)xx + (logH)yy ≥ 0, K0> 0, (−K)max ≥ (|H|max), ∂H∂x + ∂H∂y ≥ 0 and | ω | is bounded, then either (|H|max)2 + (−K)max = (H0)2 + K0 or ((|H|max)2 + (−K)max)(H2{x ∈

Key concepts: Mean curvature, Center of curvature, Curvature, Gaussian curvature, Constant-mean-curvature surface, Constant (computer programming), Geometry, Mean curvature flow

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