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Additional resources for Harmonic Mappings and Minimal Immersion, 1st Edition
16'). , since these We on first relations (o,t<) note that are true ii = I IIU,I for t = t and o , and = 0 U , are p a r a l l e l . 18) = s k +s and ~ _ s " k + sek > k + ~ +es = 0 into 2 + (s+es) account. 15) . 3. 2 is p r o v e d . . . is d e c r e a s i n g of Theorem . 1 . (O,t e) yields ff}
Can be e x t e n d e d Remark . following U :~ -Z p principle proof. 6. relatively ~ closed functions field. 5 domain caP2(Z) f 6C 3 i = n+1, on ... 5' IRn , and . , more ~ ,n+m ... general, planes plane = O .... 1, surfaces subset X = of derived n a ~) zi = describe have theorem a of that x 6 ~ -Z sufficient. 34) = O and curvature differ . More too precisely, 41 are s a t i s f i e d where fn+1 (x) . . . 42) . 32) are real analytic in the functions ~ . 42) x 6 ~ -Z Proof. We once again infer X ~ G(n,m) is h a r m o n i c .
I closed the theorem , manifold the Clearly, Bernstein's and are ~ ~3 cases f : ~n ya8 chosen that ~ ~m as = O n = 4 15, minimal , into the u is not Grass- . 3' have not hold , .. 3 p. 36) and of , it pole known sint, with S2 solution y (cost, u(~) hemisphere constant. for m = I On the . hand, where of is = u : ~ ~ S2 , p = norlth Moreover if map S 2 = G(I,2) the B /2(p) It and function. geodesic mannian is 590-591) IRn Louiville the u(t) is on unable the pp. Almgren, replacing are fact, admitted (, lSf(x) I , or In seen theorem De ~ report.