2018/12/09 by Armin Schikorra, Schikorra, Armin
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1812.03494
openalex publication_date 2018/12/09 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We consider limits of weakly converging W1,2-maps \Φk from a ball\nB \⊂ \ℝ2 into \ℝ3 which are conformal immersions.\nUnder the assumption that a normal curvature term is small, namely if for the\nnormal map u we have for some s \∈ (\(1)/(2),1)\n
intB
intB
left |
fracuk(x)
wedge uk(y)|x-y|s\n
right|^
frac2s
,
fracdx
, dy|x-y|2 lt;
varepsilon then we show\nthat we can either pass to the limit and obtain an almost everywhere immersion\n\Φ or \Φ collapses and is constant. This is in the spirit of the\nresults by T. Toro, and S. M "uller and V. Sverak, and F. H 'elein, who\nobtained similar statements under the stronger assumptions that the second\nfundamental form is bounded (but also stronger result: a locally bi-Lipschitz\nparametrization).\n The fractional normal curvature assumption is vaguely reminiscent of\ncurvature energies such as the scaling-invariant limits of tangent-point\nenergies for surfaces as considered by Strzelecki, von der Mosel et al. and we\nhope that eventually the analysis in this work can be used to define weak\nimmersions with these kind of energy bounds.\n