By Michael Atiyah

Those notes are in accordance with the process lectures I gave at Harvard within the fall of 1964. They represent a self-contained account of vector bundles and K-theory assuming basically the rudiments of point-set topology and linear algebra. one of many positive factors of the remedy is that no need is made from usual homology or cohomology thought. in truth, rational cohomology is outlined by way of K-theory.

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Fenn. Ser. A I Math. 425, 1–10 (1968) 24. : Integral means of univalent harmonic maps. Ann. Univ. Mariae Curie-Sklodwska 50, 155–162 (1996) 25. : On the accumulation of the zeros of a Blaschke product at a boundary point. Proc. Am. Math. Soc. 34, 489–496 (1972) 26. : On the fourier series of a step function. Mich. Math. J. 36, 459–475 (1989) 27. : Potential Theory in Modern Function Theory, 2nd Edn. Chelsea Publishing Company, White River (1975) 28. : On the dilatation of univalent harmonic mappings.

1 Z. Boyd and M. Dorff Background In order to explore minimal surfaces more fully, we introduce three important concepts from differential geometry, which is the study of differentiable surfaces in space. For more details on the material from this section, [7]. A surface, M ∈ R3 , can be parametrized by a smooth function x : D → R3 if x(D) = M and x is one-to-one. Parameterizing a surface with smooth functions allows us to do calculus with the surface and gives us a way to translate geometric concepts into rigorous analytic language.

3⎦ 3 The Minimal Surfaces Over the Slanted Half-Planes, Vertical Strips and Single Slit Case 3 For γ = 5π , 4 ⎤ ⎡ −iπ 4 − iπ 4 1 1 e i z−e ⎢1 F (z) = ∓Re ⎣ log + + iπ − iπ − iπ 2 2 4 z+e 4 z+e 4 z + e− 4 Case 4 For γ = ⎥ 2⎦ + c. 7π , 4 ⎡ ⎤ iπ 4 1 z+e 1 1 iπ i ⎢ 1 F (z) = ∓Re ⎣− log + e4 + iπ iπ iπ 4 2 2 z−e 4 z−e 4 z−e 4 Case 5 Let γ ∈ / 53 π 3π 5π 7π , 4, 4, 4 4 ⎥ 2⎦ + c. 4) has simple poles at ieiγ and −ieiγ , and a pole of order 2 at e−iγ . Thus, we may rewrite h (z) as h (z) = A B C D + + + z + ieiγ z − ieiγ z − e−iγ z − e−iγ 2 , where A, B, C, and D can be easily computed using a standard procedure from residue calculus or otherwise.