Abstract:

We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\{1\\over 2\}$ and arbitrary genus in $\\Bbb\{S\}^2\\times\\Bbb\{R\}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\{1\\over 2\}$ tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in $\\Bbb\{S\}^2\\times\\Bbb\{R\}$, $\\Bbb\{H\}^2\\times\\Bbb\{R\}$, and $\\Bbb\{R\}^3$ with bounded height and enjoying the symmetries of certain tessellations of $\\Bbb\{S\}^2$, $\\Bbb\{H\}^2$, and $\\Bbb\{R\}^2$ by regular polygons.

摘要：

我们获得具有恒定平均曲率$ 0 <H <\ {1 \\ over 2 \} $且在$ \\ Bbb \ {S \} ^ 2 \\ times \\ Bbb \ {R \}中具有任意属的紧凑可定向嵌入式曲面$。这些表面具有二面角对称性，并沿赤道将一对平均曲率为\\ {1 \\ over 2 \} $切线的球体异化。这是具有恒定平均曲率的$ \\ Bbb \ {S \} ^ 2 \\ times \\ Bbb \ {R \} $，$ \\ Bbb \ {H \} ^ 2 \\ times \\ Bbb \ {R \} $和$ \\ Bbb \ {R \} ^ 3 $，且高度有限，并且具有$ \\ Bbb \ {S \}某些镶嵌的对称性^ 2 $，$ \\ Bbb \ {H \} ^ 2 $和$ \\ Bbb \ {R \} ^ 2 $。