By David Mumford, C. P. Ramanujam, Yuri Manin

Now again in print, the revised version of this well known examine offers a scientific account of the elemental effects approximately abelian types. Mumford describes the analytic equipment and effects acceptable whilst the floor box ok is the advanced box C and discusses the scheme-theoretic equipment and effects used to accommodate inseparable isogenies whilst the floor box ok has attribute p. the writer additionally offers a self-contained facts of the lifestyles of a twin abeilan kind, stories the constitution of the hoop of endormorphisms, and contains in appendices "The Theorem of Tate" and the "Mordell-Weil Thorem." this can be a longtime paintings by way of an eminent mathematician and the one publication in this topic.

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N. Statement: Then there is a unique immersion X ∈ C3 (B, Rn+2 ) with an unique ONF N = (N1 , . . , Nn ) such that ˚ X(0, 0) = X, Xui (0, 0) = Z˚(i) , Nσ (0, 0) = N˚ σ . t. to the ONF N, and the Tσϑ,i ≡ Tσϑ,i represent the respective torsion coefficients. Our proof of this theorem follows the lines of Blaschke and Leichtweiss [15]. Preceded is the following lemma (see also ibidem, § 60). 1. Consider the initial value problem m ∂ zk = ∑ aℓki zℓ , i ∂u ℓ=1 zk (u0 , v0 ) = z˚k , i = 1, 2, k = 1, .

6 The Weingarten equations 31 Proof. With unknown functions aσ ,i and bσϑ ,i we evaluate the ansatz Nσ ,ui = 2 n k=1 ϑ =1 ∑ akσ ,iXuk + ∑ bσϑ ,i Nϑ . Multiplication by Xuℓ gives −Lσ ,iℓ = Nσ ,ui · Xuℓ = 2 2 k=1 k=1 ∑ akσ ,i Xuk · Xuℓ = ∑ akσ ,igkℓ , and rearranging yields 2 ℓm am σ ,i = − ∑ Lσ ,iℓ g . ℓ=1 A further multiplication by Nω shows Tσω,i = Nσ ,ui · Nω = n n ϑ =1 ϑ =1 ∑ bσϑ ,i Nϑ · Nω = ∑ bσϑ ,iδϑ ω = bσω,i , which proves the statement. 11) generalizes the classical Weingarten equations 2 Nui = − ∑ Li j g jk Xuk , i = 1, 2, j,k=1 for the unit normal vector N of a surface X : B → Rn+2 in the case of one codimension n = 1, found by the German mathematician Julius Weingarten (*1836 in Berlin; †1910 in Freiburg).

There is an endless list of contemporary studies on constant mean curvature surfaces. The reader finds various excellent contributions in the works of U. Abresch, B. Ammann, C. Gerhardt, K. Grosse-Brauckmann, F. Helein, J. Isenberg, H. Karcher, M. Kilian, K. Kenmotsu, N. Kapouleas, R. Lopez, R. Kusner, F. H. Meeks, F. Pedit, K. Polthier, N. Schmidt, J. Sullivan, M. Weber, H. Wente etc. In 1972, David Hoffman in [92] considered the embedding problem for compact surfaces with parallel mean curvature vector in four-dimensional Euclidean space.