Abstract:Let R be an open Riemann surface. In this paper we prove that every continuous function M → R n , n ≥ 3, defined on a divergent Jordan arc M ⊂ R can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions … Show more
Based on the Runge theorem for generalized analytic vectors proved by Goldschmidt in 1979, we provide a Mergelyan-type and a Carleman-type approximation theorems for solutions of Pascali systems.
Based on the Runge theorem for generalized analytic vectors proved by Goldschmidt in 1979, we provide a Mergelyan-type and a Carleman-type approximation theorems for solutions of Pascali systems.
Based on Runge theorem for generalized analytic vectors proved by Goldschmidt in 1979 we provide a Mergelyan-type and a Carleman-type approximation theorems for solutions of Pascali systems.
“…• The space CMI c nf (M, R n ) of complete nonflat conformal minimal immersions M → R n is dense (with respect to the compact-open topology) in the space CMI nf (M, R n ) of all nonflat conformal minimal immersions ([8, Theorem 7.1]; the case of n = 3 follows from [13,Theorem 5.6], which slightly predates the introduction of Oka theory in minimal surface theory). In more recent work, the density theorem has been strengthened to Mergelyan and Carleman approximation theorems including Weierstrass interpolation and other additional features (see [1], [10,Section 3.9], and [17]).…”
We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface M into R n , n ≥ 3. It follows that the inclusion of the space of such immersions into the space of all nonflat conformal minimal immersions is a weak homotopy equivalence. When M is of finite topological type, the inclusion is a genuine homotopy equivalence. By a parametric h-principle due to Forstnerič and Lárusson, the space of complete nonflat conformal minimal immersions therefore has the same homotopy type as the space of continuous maps from M to the punctured null quadric. Analogous results hold for holomorphic null curves M → C n and for full immersions in place of nonflat ones.
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