×

The geometry of minimal surfaces of finite genus I: Curvature estimates and quasiperiodicity. (English) Zbl 1068.53012

This paper opens up a series of subsequent publications by the authors on minimal surfaces of finite genus in \(\mathbb{R}^3\), to appear or available as preprints. Here, the properly embedded minimal surfaces \(M\) with genus zero and two limit ends are investigated, supposing that everyone of them is normalized so that they have horizontal limit tangent planes at infinity and the vertical component of their flux equals one. (Here flux is the integral along an embedded closed curve \(\delta\subset M\) of the unit tangent vector to \(M\) and orthogonal to \(\delta\).) It is proven that if a sequence \(\{M(i)\}\) of such surfaces has the horizontal part of the flux bounded from above, then the Gaussian curvature of the sequence is uniformly bounded. In the proof, results are used from forth coming papers by Golding and Minicozzi (to appear in Ann. Math.).
Then “genus zero” is replaced by “finite genus” and normalization is supposed to be oracle by a rotation and homothety. Also here some curvature estimates are obtained which yield compactness results and quasiperiodicity of \(M\) in the following sense: There exists a divergent sequence \(V(n)\in\mathbb{R}^3\) such that the translated surfaces \(M+ V(n)\) converge to a properly embedded minimal surface of genus zero with, two limit ends, horizontal limit tangent plane at infinity and with the same flux as \(M\). Also, some other applications are given.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
54C25 Embedding
PDFBibTeX XMLCite
Full Text: DOI