Limits of Conformal Immersions Under a Bound on a Fractional Normal Curvature Quantity
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Limits of Conformal Immersions Under a Bound on a Fractional Normal Curvature Quantity Armin Schikorra1 Received: 17 January 2020 / Accepted: 11 May 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We consider limits of weakly converging W 1,2 -maps k from a ball B ⊂ R2 into R3 which are conformal immersions. Under the assumption that a normal curvature term is small, namely if for the normal map u we have for some s ∈ ( 12 , 1) 2 uk (x) ∧ uk (y) s dx dy |x − y|s |x − y|2 < ε B B then we show that we can either pass to the limit and obtain an almost everywhere immersion or collapses and is constant. This is in the spirit of the results by T. Toro, and S. M¨uller and V. Sverak, and F. H´elein, who obtained similar statements under the stronger assumptions that the second fundamental form is bounded (but also a stronger result: a locally bi-Lipschitz parametrization). The fractional normal curvature assumption is vaguely reminiscent of curvature energies such as the scaling-invariant limits of tangentpoint energies for surfaces as considered by Strzelecki, von der Mosel et al. and we hope that eventually the analysis in this work can be used to define weak immersions with these kind of energy bounds. Keywords Conformal immersions · Fractional curvature Mathematics Subject Classification (2010) 53A30 · 53A05 · 35J35 · 35J60
1 Introduction Let : B(0, 1) → R3 be a conformal parametrization of a patch of a surface ⊂ (B(0, 1)). The Willmore energy of this patch is given as 1 W (; B(0, 1)) = |∇u|2 + C(), 4 B(0,1) This work is dedicated to J¨urgen Jost on the occasion of his 65th birthday. Armin Schikorra
[email protected] 1
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
A. Schikorra
where u is the unit normal to at (x) and C() is a constant depending on the topology of . ∂2 ∂1 ∧ . u(x) = |∂1 | |∂2 | ˇ ak [22] after earlier works by Toro The following is a fundamental theorem by M¨uller–Sver´ [41, 42], see also [12, Theorem 5.1.1]. Sharp constants ε0 were obtained in [15, 19]. We also refer to surveys [16] and [31]. Theorem 1.1 There exists ε0 > 0 such that the following holds. Assume that k ∈ C ∞ (B(0, 1), R3 ) is a sequence of conformal immersions, i.e. for each k ∈ N ∂α k = eλk eα;k ,
α = 1, 2
C ∞ (B(0, 1), R3 )
and λk ∈ C ∞ (B(0, 1)). for some orthonormal system e1;k , e2;k ∈ 1,2 3 If k converges weakly to in W (B(0, 1), R ) and if sup W (k ; B(0, 1)) < ε0 , k∈N
then is either a constant map or is a bilipschitz conformal immersion. 1,2 1,2 2 More precisely, there are λ ∈ L∞ loc ∩ Wloc (B(0, 1)) and (e1 , e2 ) ∈ Wloc (B(0, 1), S ) an orthonormal system such that ∂α = eλ eα ,
α = 1, 2.
This theorem has had numerous applications to define and analyze weak Willmore immersions, see the celebrated [30]. On the other hand, in recent works [3, 4] there has been a breakthrough in the regularity theory of critical points of knot energies, namely M¨obiu
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