On constant mean curvature graphs with convex boundary
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We give area and height estimates for cmc-graphs over a bounded planar C2,α domain ⊂ R3. For a constant H satisfying H2|Ω|≤ 9π/16, we show that the height h of H-graphs over with vanishing boundary satisfies |h| < (˜r/2π)H|Ω|, where ˜r is the middle zero of (x−1)(H2|Ω|(x+2)2 −9π(x−1)). We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying H2|Ω| < (297 − 13)π/8, there exists an H-graph with vanishing boundary.
Abstract
We give area and height estimates for cmc-graphs over a bounded planar C2,α domain ⊂ R3. For a constant H satisfying H2|Ω|≤ 9π/16, we show that the height h of H-graphs over with vanishing boundary satisfies |h| < (˜r/2π)H|Ω|, where ˜r is the middle zero of (x−1)(H2|Ω|(x+2)2 −9π(x−1)). We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying H2|Ω| < (297 − 13)π/8, there exists an H-graph with vanishing boundary.
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