By Katsuei Kenmotsu

ISBN-10: 0821834797

ISBN-13: 9780821834794

The suggest curvature of a floor is an extrinsic parameter measuring how the outside is curved within the three-d house. A floor whose suggest curvature is 0 at every one element is a minimum floor, and it really is recognized that such surfaces are versions for cleaning soap movie. there's a wealthy and famous thought of minimum surfaces. A floor whose suggest curvature is continuing yet nonzero is acquired after we attempt to reduce the realm of a closed floor with no altering the amount it encloses. a simple instance of a floor of continuing suggest curvature is the field. A nontrivial instance is equipped via the consistent curvature torus, whose discovery in 1984 gave a robust incentive for learning such surfaces. Later, many examples of continuing suggest curvature surfaces have been came upon utilizing numerous equipment of study, differential geometry, and differential equations. it truly is now turning into transparent that there's a wealthy conception of surfaces of continuing suggest curvature.

In this booklet, the writer provides a number of examples of continuing suggest curvature surfaces and strategies for learning them. Many finely rendered figures illustrate the implications and make allowance the reader to imagine and higher comprehend those attractive items.

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W hen a surface X(D) is embedded and does not contain the origin of R 3, the absolute value of \CX\ is the volume of the cone CX over X(D). Hence, if X(D) is a closed convex surface, the origin of R 3 lies inside the surface, and the outward normal vector is taken as the normal vector, then \CX\ is the volume of the domain enclosed by X(D). 7). R emark . (1) Let X(u,v\t) be a variation of X(u,v) satisfying X ( u ,v ;0 ) = X(u, u), dX(u, v\ t) dt t=o Y{u,v). Considering the Taylor expansion of X(u, v;t) with respect to t, we can see that the variation X t(u,v) considered in this section is the linear approximation of X(u,v; t).

P roof. We obtain H = 0 by R — ►oo in Theorem 2. 1. 2 follows. | R emark . Since the mean curvature of the graph of a function f(u,v) = VR2 —u2 —v2 (u2 + v2 ^ R2) is constant and H = 1/i ? 1 cannot be improved. 2. The area functional The area element of a local parametric surface X : D — ►R 3 is dA = y/EG —F 2 dudv, and the area of X is given by \X\D = [ dA. Jd We will call this the area functional. 2. THE AREA FUNCTIONAL 27 Let D ( c R 2) be a relatively compact domain in R 2 whose bound­ ary is smooth.

3). 4) Z\s) - 2%/ = l H(s)Z{s) - 1 - 0, s £ I. 6) Z(s) = { ( F(s) - Cl) + V=L ( G(s) + c2) } ( F '( s ) - G '(s)), where c = V ^ l (ci — -\/—IC 2) is an arbitrary complex constant. The function Z(s) is defined in terms of functions x(s), y(s), and their derivatives. Conversely, we can obtain y(s) and x(s) from Z(s). In­ deed, since \Z(s)\2 = y(s)2, we have V(a) = V(F( s) ~ c1)2 + (G(s) + c2)2, s € /, and since Z(s) - Z(s) = 2v/ = lj/( s ) x , (s), we have _ ( ) (G (s) + c2)F'( s) - ( F ( s) - Ci )G'(s) y/(F(s) - c i)2 + (G ( s ) + c2 )2 „^ r ’ Putting these calculations together, the following theorem is ob­ tained.

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Surfaces With Constant Mean Curvature by Katsuei Kenmotsu


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