On the existence of a harmonic mapping

Lemma 1. The family Ω is normal in each singular simplex of 2.

Lemma 2. It is possible to introduce the distance between two points on R by means of any ω belonging to Ω,.

Proposition 1. is non-void.

Lemma 3 § ω q

Theorem 1. The family %ω is normal on R. oo

Proposition 2. fooip) is schlicht.

Lemma 4. The sequence {fn 1} converges uniformly on R'm oo

Proposition 3. fz 1 is continuous.

Proposition 4A. /«, is U-deriυable and measurable (i.e., absolutely continuous in 2-dimensional sense) on R.

Proposition 4B. f' 1 is U-derivable and measurable (i.e., absolutely continuous in 2-dimensional sense) on R'.

Proposition 5 is homotopic to X.

Proposition 6. f^ is a sense-preserving map.

Proposition 7.

Proposition 8. There exists at least one homeomorphism fω G Sω > such Formulation and solution of the second extremum problem Consider the family i$Q={fω, ®GΩ} consisting of all the solutions of our first problem. Clearly SαΦΦ Now we shall settle the second extremum problem as follows: Minimize the functional within the family f?Q, wω(z) being a local realization of fω. oo By definition we can find a minimizing sequence {/„„} such that Lemma l' %Q is equicontinuous. ' oo$

Theorem 1'. The family %Q is normal on R.

Lemma 5. The family/ O is compact on R. oo

Lemma 6. Let pj 0 = 1, 2) δ# arbitrary points on a singular simplex Si and let Γ the class of all rectifiable simple arcs which connects the two points pj on S7. Then for any ye Γ and for any ω = τ(z)dze ί2, the quantity Jl«l = J 1 ^ ) 1 1 ^ 1 y y is bounded away from zero.

Proposition 9. g^p) is schlicht.

Proposition 10. goo(P) is a homeomorphism between R and R'.

Proposition 11A. g^P) is U-deriυable and absolutely continuous in 2-dirnensional sense on R.

Proposition 11B. gZ 1^) is U-derivable and absolutely continuous in 2-dimensional sense on R'. The proofs are similar to that of Proposition 4A (resp. Proposition 4B), since g^ (resp. gz 1) is the uniform limit of {fω } (resp. {/']} ).

Proposition 12. g^ is homotopic to %.

Proposition 13 g^ is a sense-preserving map. The proofs are similar to that of Propositions 5 and 6, since g^ is oo the uniform limit of {fωj . The class O of the locally holomorphic differentials ω is compact by oo

Lemma 5, so the sequence {ωM} contains at least one subsequence uni- formly convergent in R— U 3S, , the limit of which we denote by ω^ = τ00(z)dz; it belongs again to ίλ Let zωn(w), Z^iw) be local realizations °f fωltiΨy g^fa) respectively. Then, using the similar reasoning to that in the proof of Proposition 7, we see dw inf( B 2 ( ( J V 2)h 2 r.(Z.( \dwΛdW\ <M-(K+^) .

Proposition 14. The Propositions 9, 10, 11A, 11B, 12, 13 and 14 imply the fact that #«, belongs to Sωoo. We are now going to compare it with /ωoo, defined oo as one of the extremal maps in %ω since {fωΛ converges to g^ uni- formly on i?, we have by the procedure often employed (cf. Proposition 7) (19) -^ 1™ \\ 3z 2 + dz

Proposition 15. // M is sufficiently large, there exists at least one locally holomorphic differential ώM£ί2M, such that (20)

Proposition 15 shows the fact that the map fM belongs to the family %ωMy if M is sufficiently great. This mapping, by definition, extremizes the functional /[/] in the wider family %gM much more does in S^M- Now we are in a position to get the necessary condition for the map / = / M to minimize the Dirichlet functional within the family SωM To this end, we set a = pdz + qdz = p(w(z))(dz + with a local realization w(z) of Let F be a compact set comprised in a local coordinate neighbourhood ufty (

Theorem 2. There exists at least one topological mapping from R to R / which belongs to the given homotopy class and is harmonic relative to the given con formal metric on R'. EXAMPLE. Consider the simplest case in which R and R f are both tori (η, A being naturally fixed). Then the harmonic map of R onto R / is unique except the conformal mappings of R onto itself. Suppose there exist two such mappings, say, / and /. Let q be an arbitrary point on R' and af9 ay the holomorphic quadratic differe