 By W. Browder

Brower W. surgical procedure on simply-connected manifolds (Springer, 1972)(ISBN 0387056297)

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Additional resources for Surgery on simply-connected manifolds

Sample text

55) family given by the intrinsic isometry w- e'aw, Proof. The aER. 46). The right hand side obviously harmonic. Integrating the harmonicity condition for the left hand for is side one obtains h" W = h' Inserting this into C, h + C2, C, E R. 59). Any solution of -1. 46). h'(w) h(w) 2 2 p E R. 52) to the form J+2) (I+WJ+2). 56). 4. 58). 2. 1. 2. Let F be critical point Proof. w = 0 Let us we can of the mean an Theory immersion of a new w as surface Y C R' with surface of type B. Bonnet a curvature.

3 shows a Bonnet surface (fainily) of type B. Again the tubes are trajectories of the isometric flow preserving the mean curvature function. 6 Fig. 3. A branched Bonnet surface Fig. 4. Bonnet surface of immersion domain U of Bonnet surfaces of type B is mental domains U,, Indeed, 55 = JW E the fundamental forms C1 (n (see - naturally split type B into funda- 7r 1)7r2 < Im(w) Table 3. 1) are < n 2 invariant with. respect to the shift 7r i 2 are congruent in W. 3. For an appropriate choice of parameters, several copies of the fundamental domain can close up and thus comprise a closed surface with a critical point.

42) z a = JW12 in this local chart6 H(s). 2 coordinate s. we keep W, (z) = i W(z) f (z). 43) H(S), but for another 3. 4. Consider (3-42). Cv = reads as Hf, the claim. 41) in terms of w, holomorphic. Thus, w which '9 W f az > 0. 44) satisfies AH'(s)<0,VsE(0,E). 47) T_(w)) + h(w))2 Idw 12. 47). 48). , c), and tal forms with H(s), Q(tv,iT), e` ( "Y"" dideraline a Bonnet surface in RI with J (isolated) critical point w 0 of ' 1, =- Proof. 37) implies h(w). 46). 36), (3-43). 48). 44). 45). 46). 48) together satisfy the Codazzi equations for w 0 0 when the mean curvature function is a function of s JW12 only.