C^{1,\alpha }-regularity for surfaces with H\in L^p |
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Authors: | Theodora Bourni Giuseppe Tinaglia |
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Institution: | 1. Institute of Mathematics, Free University Berlin, Arnimallee 7, 14195?, Berlin, Germany 2. Department of Mathematics, King’s College London, The Strand, London?, WC2R 2LS, UK
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Abstract: | In this paper, we prove several results on the geometry of surfaces immersed in \(\mathbb {R}^3\) with small or bounded \(L^2\) norm of \(|A|\) . For instance, we prove that if the \(L^2\) norm of \(|A|\) and the \(L^p\) norm of \(H\) , \(p>2\) , are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded \(L^2\) norm of \(|A|\) , not necessarily small, then such a disk is graphical away from its boundary, provided that the \(L^p\) norm of \(H\) is sufficiently small, \(p>2\) . These results are related to previous work of Schoen–Simon (Surfaces with quasiconformal Gauss map. Princeton University Press, Princeton, vol 103, pp 127–146, 1983) and Colding–Minicozzi (Ann Math 160:69–92, 2004). |
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