Minimal Surfaces I: Boundary Value Problems by Ulrich Dierkes

By Ulrich Dierkes

Minimum surfaces I is an advent to the sector ofminimal surfaces and apresentation of the classical theoryas good as of elements of the fashionable improvement centeredaround boundary price difficulties. half II bargains with theboundary behaviour of minimum surfaces. half I isparticularly apt for college students who are looking to input thisinteresting quarter of research and differential geometry whichduring the final 25 years of mathematical examine has beenvery energetic and effective. Surveys of assorted subareas willlead the coed to the present frontiers of data andcan alsobe invaluable to the researcher. The lecturer caneasily base classes of 1 or semesters on differentialgeometry on Vol. 1, as many themes are labored out in greatdetail. a number of computer-generated illustrations of previous andnew minimum surfaces are integrated to aid instinct andimagination. half 2 leads the reader as much as the regularitytheory fornonlinear elliptic boundary price problemsillustrated by way of a specific and interesting subject. There isno comparably accomplished therapy of the matter ofboundary regularity of minimum surfaces to be had in bookform. This long-awaited booklet is a well timed and welcomeaddition to the mathematical literature.

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The Weingarten equations (38) imply that Nul whence (44) A Nu2 = (bI bi - bl bt)Xul A Xu2 20 1. o. o. Then AQ,(X) = f n, IXu /\ Xvldudv and therefore An,(N) _ dAN ( )1 -- l'I m IKwo ---- . 0 is the ratio of the area elements dAN and dAx of the "spherical image" N of X and of X itself. 0. On account of (44), the surface normal N = INu /\ Nvl-1(Nu /\ N v ) will there be well defined, and ° (46) N= Nif K > 0, N= -NifK 0, but (b) orientation reversing if K < 0.

3, we have Kgds = Kgfldt = (1 + :v log fl)dt, where v is the exterior normal of oQ. By virtue of Gauss's integral theorem, we thus obtain -f x KdA = f ,1 log fldudv = f21[ = L21[ (Kgfl- l)dt = ~ov log fldt L 0 Q Kgds - 2n which proves (10). Ix Next we note that K dA remains the same if X is replaced by a surface X 0 T, where T: Q* -+ Q is a C 1 -diffeomorphism, and similarly SrKgds is unchanged if we assume the Jacobian of T to be positive. Hence the left hand side of (10) is an invariant of X 0 T with respect to all parameter changes by diffeomorphisms T E C 1 (Q*, [R2) with Q = T(Q*) and J t > O.

A letter to Schumacher dated July 5, 1816 documents that the solution of this problem was known to Gauss, and a note of this solution is preserved (cf. Werke 8 [2], p. 371). " When in 1821 no solution came in, the question was renewed for 1822. Having been urged by Schumacher, Gauss sent his contribution to Copenhagen on December 11,1822, and, in 1823, he obtained the prize of the academy. In 1825, Gauss's paper was published in the last issue of Schumacher's "Astronomische Abhandlungen". The main result ofthis investigation is that every sufficiently small piece of a regular, real-analytic surface can be mapped cmiformally onto a domain in the plane.

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