By Jean Dieudonné

Written by means of a world-renowned mathematician, this good educated and vintage textual content lines the heritage of algebraic topology, starting with its construction within the early 1900s. It is going directly to describe intimately the $64000 theories that have been stumbled on prior to 1960.

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3. Continuity o f weak solutions 37 We shall again exclude that 6 assumes a maximum on Q u Di to prove the result. 2) (2. 2. 8) and similarly for Likewise and q>2 . 9) and again similarly for cp^ and cp2 . 2. 10) with i c > K . _ Since grad0 is always trangential to Di because of property (i), 0 cannot assume its maximum on Q u D i. d. We shall also need the global uniqueness theorem of Al’ber and Hartman (see [Albl], [Alb2], [Ht], [SY3]; cf. 8] for a proof). 3: Let u:M -^N be a harmonic map between compact Riemannian manifolds {without boundary).

D j = l + l ...... d. 18) holds for this basis, and putting for J = / + 1,.. 18) <^o)b‘l hence also 1 —c For simplicity, we shall omit the symbol for the rest of the argument. In particular, for small enough (Tq, the v‘ are linearly independent. , d and we have to estimate |Bi; —u| for arbitrary v. 19) already gives the necessary control of D^. 17). ,d -l) K - o)DvM c'V',€T^(H)N. 3. C ontinuity o f weak solutions 49 Therefore, in any case, for veT^N with /ig M((7o) - vPhn^v\ < C2ifi, A , Сто,d).

5) (of course, as before + g„j - gj^_,)). e. intrinsic quantities of the surface. t. the first fundamental form. 6) w with W^:= E G -F ^ = \x„ A x„\^. Since n is always of length 1, n = n{u,v) defines a map into S^, the spherical image or Gauss map of x (m, v). In the (u, v) parameters, the metric on then becomes dn^ = du^ + 2n„n„du dv + dv^. e. parameters (m, u) for which E=G F = 0. 6) become L M П и = - - Х и - — х„ E E M N n „ = --x „ --x „ . 14) The following is a special case of a result of Ruh and Vilms [RV].