Correction to: Interpolation and optimal hitting for complete minimal surfaces with finite total curvature
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Calculus of Variations
CORRECTION
Correction to: Interpolation and optimal hitting for complete minimal surfaces with finite total curvature Antonio Alarcón1 · Ildefonso Castro-Infantes2 · Francisco J. López1 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Correction to: Calc. Var. (2019) 58:21 https://doi.org/10.1007/s00526-018-1465-0 Throughout this note we use the notation from [1]. Franc Forstneriˇc pointed out that [1, Proposition 2.3] is incorrect by giving us a counterexample; we are grateful to him for reporting and for helpful discussions. We overlooked the possible contribution of the cusp points of the curve γ in the result to the absolute value of its winding number. To fix this, we provide a corrected version of [1, Proposition 2.3] which measures the possible influence of the cusp points on the winding number of the curve (see Proposition 2.3). We then use this result to obtain corrected versions of [1, Theorems 1.3 and 1.4] on optimal hitting theory for complete minimal surfaces (see Theorems 1.3 and 1.4). The other main results in [1], namely [1, Theorems 1.1 and 3.1], are not affected by the error. Here are the corrected statements; see [1, Sects. 1 and 2.5] for background. Proposition 2.3 Let γ (t) be a real analytic, piecewise regular, closed curve in R2 admitting a regular normal field. Denote by tγ and m γ the turning number and the number of cusp points of γ , respectively. Fix p ∈ R2 \ γ and let wγ ( p) denote the winding number of γ with respect to p. Then, we have |wγ ( p)| ≤ 2tγ + 21 m γ . The addend 21 m γ is missing in the bound stated in [1, Proposition 2.3]. Theorem 1.3 Let r ≥ 1 be an integer. For any integer m with 2 − 2r ≤ m ≤ 1 there is a set Ar ;m ⊂ R3 which is against the family k≤m Zr ;k and consists of 12r +50r 3 +2m +1 points
The original article can be found online at https://doi.org/10.1007/s00526-018-1465-0.
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Antonio Alarcón [email protected] Ildefonso Castro-Infantes [email protected] Francisco J. López [email protected]
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Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, 18071 Granada, Spain
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Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain 0123456789().: V,-vol
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whose affine span is a plane. In particular, the set Ar ;1 , which consists of 12r + 50r 3 + 3 points, is against the family Zr . Thus, if X : M → R3 is a complete nonflat orientable immersed minimal surface with empty boundary and the Euler characteristic χ(M) ≤ m, and if Ar ;m ⊂ X (M), then the total curvature TC(X ) < −4πr . In particular, no complete nonflat orientable immersed minimal surface X with |TC(X )| ≤ 4πr contains Ar ;1 . Theorem 1.4 Let X : M → R3 be a complete orientable immersed minimal surface of finite total curvature and empty boundary. If L ⊂ R3 is a straight line not contained in X (M), then # X −1 (L) ≤ 6Deg(N ) + 25Deg(N )3 + χ(M) where Deg(N ) is the degree of the Gauss map N of X and χ(M) is the
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