Differential Geometry

Read e-book online Elements of the geometry and topology of minimal surfaces in PDF

By A T Fomenko and Avgustin Tuzhilin, Visit Amazon's A.T. Fomenko Page, search results, Learn about Author Central, A.T. Fomenko, , Avgustin Tuzhilin

ISBN-10: 0821837915

ISBN-13: 9780821837917

ISBN-10: 0821845527

ISBN-13: 9780821845523

This e-book grew out of lectures offered to scholars of arithmetic, physics, and mechanics by means of A. T. Fomenko at Moscow collage, lower than the auspices of the Moscow Mathematical Society. The ebook describes sleek and visible facets of the conception of minimum, two-dimensional surfaces in third-dimensional area. the most themes lined are: topological houses of minimum surfaces, good and volatile minimum movies, classical examples, the Morse-Smale index of minimum two-surfaces in Euclidean house, and minimum motion pictures in Lobachevskian area. Requiring just a usual first-year calculus and undemanding notions of geometry, this ebook brings the reader swiftly into this attention-grabbing department of contemporary geometry.

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Additional info for Elements of the geometry and topology of minimal surfaces in three-dimensional space

Example text

We shall prove by contradiction that there is no closed minimal surface in R3 . Let M be a closed minimal surface in R3 . Then since M is bounded as a subset of R3 there is a plane that does not intersect M. We move this plane parallel to itself in the direction of M until it first touches M at some point P (we denote the tangent plane by n). Clearly, P is an internal point of M and the plane II (M has no boundary points because it is closed) and M lies on one side of lI . By the maximum principle M is flat in some neighborhood of P.

However, in geometry there is another definition of completeness of a surface, imposing weaker restrictions, which nevertheless extends to a wider class of surfaces and turns out to be quite sufficient to obtain meaningful results. DEFINITION. Let M be an arbitrary immersed connected surface, and p the intrinsic metric defined above. Then M is said to be complete if the metric space (M, p) is complete. If M is an embedded connected surface, then it is easy to see that p(A, B) > IABI for any points A and B of M.

In this coordinate system the surface M in a neighborhood of P can be given by a graph x = x, y = y, z = f (x , y). All such functions f are described by the minimal surface equation (see § 1 of Chapter 2): (I+Jy)fxx-2fxfyfxy+(1+ f2)fyy=0. Let us form three functions: 1+f 1+ f2+fy' 1+f ffy 1+ f2+ fY 1+f2+fy' Henceforth for convenience we put p = fx , q = f, , and w = J1 + f2 + fy . We recall that V l + f2 + f? dx d y is the element of area of the surface M . It turns out that from the minimal surface equation we have the following relations: (I wp2) y- (wq)x' (1 wg2)x= (wq)r In fact, direct calculation shows that Y (I x Ulfy) x - (f fy) y = - wx [(l + Jy )fxx - 2fj fxy + (1 + f) f y].

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Elements of the geometry and topology of minimal surfaces in three-dimensional space by A T Fomenko and Avgustin Tuzhilin, Visit Amazon's A.T. Fomenko Page, search results, Learn about Author Central, A.T. Fomenko, , Avgustin Tuzhilin


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