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By Tobias Holck Colding, William P. Minicozzi II

ISBN-10: 0821853236

ISBN-13: 9780821853238

Minimum surfaces date again to Euler and Lagrange and the start of the calculus of diversifications. the various thoughts built have performed key roles in geometry and partial differential equations. Examples contain monotonicity and tangent cone research originating within the regularity conception for minimum surfaces, estimates for nonlinear equations in line with the utmost precept coming up in Bernstein's classical paintings, or even Lebesgue's definition of the vital that he constructed in his thesis at the Plateau challenge for minimum surfaces. This booklet begins with the classical thought of minimum surfaces and finally ends up with present study themes. Of a few of the methods of forthcoming minimum surfaces (from advanced research, PDE, or geometric degree theory), the authors have selected to target the PDE points of the speculation. The ebook additionally comprises a number of the functions of minimum surfaces to different fields together with low dimensional topology, basic relativity, and fabrics technology. the one necessities wanted for this ebook are a simple wisdom of Riemannian geometry and a few familiarity with the utmost precept

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Extra resources for A course in minimal surfaces

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X, y) - 6 Gib einen anderen Beweis von Proposition 1, indem Du Proposition 2 auf h (x, y, z) anwendest. 7 Es sei [(x, y, z) = (x + y + Z _1)2. a) Bestimme die kritischen Punkte von f. b) Fiir welche c ist die Menge [(x, y, z) =c eine regulare Flache? c) Beantworte die Fragen aus a und b flir die Funktion [(x, y, z) =xyz2. 8 Es sei x(u, u) wie in Definition 1. \ dXq: 1t2 - 1t3 genau dann injektiv ist, wenn ~x z au ax av -r- 0• A -4- 1\ 9 Es sei V eine offene TeUmenge der xy-Ebene. Zeige dd die Menge {(x, y, z) E 1R3; Z = 0 und (x, y) E V} eine regulare Flache ist.

Man sieht leicht ein, d~ der Torus durch drei derartige Koordinatenumgebungen tiberdeckt werden kann. \ der Zylinder {(x, y, z) E 1t3; x2 + y2 = I} eine reguliire Flache ist, und gib Parametrisierungen an, deren Koordinatenumgebungen ihn iiberdecken. ;; 11 eine regulare Flache? Ist die Menge {(x, y, z) E 1t3; Z =0 und x 2 + y2 < I} eine reguliire Flache? h. die Menge {(x, y, z) E 1t3; x 2 + y2 - z2 =O}, keine reguliire Flache ist. 4 Es sei [(x, y, z) = z2. \ 0 kein regularer Wert von [ist, [-1 (0) aber dennoch eine reguliire Flache.

Der Normalenvektor unstetig ist bei t = 0, weswegen wir Punkte mit k = 0 ausgeschlossen haben). d) Zeige: T kann so definiert werden, daJl. T 0 ist, obwohl 0< keine ebene Kurve ist. = 11 Oft sind ebene Kurven in Polarkoordinaten durch p = p(O), a) Zeige, daJl. ;;; b, gegeben. + (p')2 dO, ist, wo der Strich die Ableitung nach 0 bedeutet. b) Zeige, daJl. die Kriimm ung k(O) = 2(p')2 _ Pp" + p2 {(p')2 + p2}l/2 ist. 12 Es sei 0<: 1-> 1R3 eine regulare parametrisierte Kurve (nicht notwendig nach der Bogenlange) und (3: J -> 1R 3 eine Umparametrisierung von o«l) nach der Bogenlange S = set), gemessen von to E I (siehe Bern.

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A course in minimal surfaces by Tobias Holck Colding, William P. Minicozzi II


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