Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON CONSTANT MEAN CURVATURE GRAPHS WITH CONVEX BOUNDARY

  • Park, Sung-Ho (Major in Mathematics Graduate School of Education Hankuk University of Foreign Studies)
  • Received : 2012.08.13
  • Published : 2013.07.31
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Abstract

We give area and height estimates for cmc-graphs over a bounded planar $C^{2,{\alpha}}$ domain ${\Omega}{\subset}\mathbb{R}^3$. For a constant H satisfying $H^2{\mid}{\Omega}{\mid}{\leq}9{\pi}/16$, we show that the height $h$ of H-graphs over ${\Omega}$ with vanishing boundary satisfies ${\mid}h{\mid}$ < $(\tilde{r}/2{\pi})H{\mid}{\Omega}{\mid}$, where $\tilde{r}$ is the middle zero of $(x-1)(H^2{\mid}{\Omega}{\mid}(x+2)^2-9{\pi}(x-1))$. We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying $H^2{\mid}{\Omega}{\mid}$ < $(\sqrt{297}-13){\pi}/8$, there exists an H-graph with vanishing boundary.

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Acknowledgement

Supported by : Hankuk University of Foreign Studies

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