Complete bounded null curves immersed in $${\mathbb {C}^3}$$ and $${\rm {SL}(2,\mathbb {C})}$$

Martin, Francisco; Umehara, Masaaki; Yamada, Kotaro
September 2009
Calculus of Variations & Partial Differential Equations;Sep2009, Vol. 36 Issue 1, p119
Academic Journal
We construct a simply connected complete bounded mean curvature one surface in the hyperbolic 3-space $${\mathcal {H}^3}$$. Such a surface in $${\mathcal {H}^3}$$ can be lifted as a complete bounded null curve in $${\rm {SL}(2,\mathbb {C})}$$. Using a transformation between null curves in $${\mathbb {C}^3}$$ and null curves in $${\rm {SL}(2,\mathbb {C})}$$, we are able to produce the first examples of complete bounded null curves in $${\mathbb {C}^3}$$. As an application, we can show the existence of a complete bounded minimal surface in $${\mathbb {R}^3}$$ whose conjugate minimal surface is also bounded. Moreover, we can show the existence of a complete bounded immersed complex submanifold in $${\mathbb {C}^2}$$.


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