6533b854fe1ef96bd12af1eb

RESEARCH PRODUCT

Normal Coulomb Frames in $${\mathbb{R}}^{4}$$

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Now we consider two-dimensional surfaces immersed in Euclidean spaces \({\mathbb{R}}^{n+2}\) of arbitrary dimension. The construction of normal Coulomb frames turns out to be more intricate and requires a profound analysis of nonlinear elliptic systems in two variables. The Euler–Lagrange equations of the functional of total torsion are identified as non-linear elliptic systems with quadratic growth in the gradient, and, more exactly, the nonlinearity in the gradient is of so-called curl-type, while the Euler–Lagrange equations appear in a div-curl-form. We discuss the interplay between curvatures of the normal bundles and torsion properties of normal Coulomb frames. It turns out that such frames are free of torsion if and only if the normal bundle is flat. Existence of normal Coulomb frames is then established by solving a variational problem in a weak sense using ideas of F. Helein (Harmonic Maps, Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002). This, of course, ensures minimality, but we are also interested in classical regularity of our frames. For this purpose we employ deep results of the theory of nonlinear elliptic systems of div-curl-type and benefit from the work of many authors: E. Heinz, S. Hildebrandt, F. Helein, F. Muller, S. Muller, T. Riviere, F. Sauvigny, A. Schikorra, E.M. Stein, F. Tomi, H.C. Wente, and many others.