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By Luther Pfahler Eisenhart

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Extra info for An introduction to differential geometry, with use of the tensor calculus

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E. annehmen, daß für z E U' Xl, x 2 -Ebene darstellen, f(z) = (z, cp(z)) mit einer Funktion cp : U' ~ lR. Wir können auch annehmen, daß U' glatten Rand hat. Wir setzen nun das Normalenfeld n von.! auf U' x lR. durch n(z, t) := n(z) auf den Zylinder U' x lR. C ]E3 fort. Man kann dies auch geometrisch dahingehend interpretieren, daß n(z, t) das Normalenvektorfeld der Fläche ft(z) = (z, f(z) + t) ist. 1 beschrieben, wählen wir nun el, e2 als Orthonormalbasis von Tf(z)f(U). Dann gilt für die Koeffizienten der ersten Fundamentalform gij(Z) = Dij und daher ist die mittlere Krümmung im Punkte f (z) 1..

Jahrhundert von Lagrange als mathematisches Problem formuliert worden war. Wir wollen das Problem folgendermaßen präzisieren: Es sei D:= {z = (x,y) E JR2, Izl 2 < I} die offene Einheitskreisscheibe, jj ihr Abschluß. Gesucht ist dann eine Abbildung 54 4. Minimalfiächen. Das Plateausche Problem mit den folgenden Eigenschaften: i) f ist in D harmonisch, also Llf(z) = 0 für alle z E D, wobei Ll hier der euklidische Laplaceoperator ~ ü) f +~ ist. ist in D konform, also < f:z;, f:z; >=< f y, fy >, < f:z;, fy >= O.

Somit gilt für den Normaleneinheitsvektor n(z) = 2Img Igl 2-1) Igl2 + 1' Igl 2+ 1' Igl 2+ 1 . ( 2Reg Dieser Ausdruck läßt sich nun auch in dem Punkte definieren, wo die Minimalfläche nicht regulär ist. Folglich ist die Gaußabbildung auch für eine parametrische Minimalfläche im Sinne von Def. 4 definiert. Übrigens ist g(z) dann auch das Bild von n(z) unter der stereographischen Projektion der Einheitssphäre 8 2 auf die Ebene C. Die meromorphe Funktion 9 ist also nichts anderes als die Gaußabbildung der Minimalfläche in stereographischer Projektion.

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An introduction to differential geometry, with use of the tensor calculus by Luther Pfahler Eisenhart

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