By Anthony Tromba

These lecture notes are in keeping with the joint paintings of the writer and Arthur Fischer on Teichmiiller idea undertaken within the years 1980-1986. because then a lot of our colleagues have inspired us to put up our method of the topic in a concise structure, simply obtainable to a vast mathematical viewers. even if, it used to be the invitation by means of the school of the ETH Ziirich to carry the ETH N achdiplom-Vorlesungen in this fabric which supplied the chance for the writer to strengthen our examine papers right into a structure compatible for mathematicians with a modest heritage in differential geometry. We additionally was hoping it is going to give you the foundation for a graduate path stressing the appliance of basic principles in geometry. For this chance the writer needs to thank Eduard Zehnder and Jiirgen Moser, appearing director and director of the Forschungsinstitut fiir Mathematik on the ETH, Gisbert Wiistholz, chargeable for the Nachdiplom Vorlesungen and the total ETH school for his or her aid and hot hospitality. This new method of Teichmiiller concept awarded right here used to be undertaken for 2 purposes. First, it used to be transparent that the classical technique, utilizing the speculation of extremal quasi-conformal mappings (in this method we thoroughly keep away from using quasi-conformal maps) was once no longer simply appropriate to the idea of minimum surfaces, a box of curiosity of the writer over a long time. moment, many different energetic mathematicians, who at a variety of occasions wanted a few Teichmiiller idea, have came upon the classical process inaccessible to them.

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Again, as above there is a unique extremal quasi conformal mapping: It is not at all obvious and needed Teichmiiller's brilliant insight to discover that extremal quasiconformal mappings can be identified with holomorphic quadratic differentials. On the other hand, from the Riemannian geometer's approach, these holomorphic quadratic differentials appear quite naturally and in a straightforward way. Much of our previous work comes to fruition in the following two theorems. 2 Given go E M there exists a local COO submanifold S of M~l (for any s) of dimension 6 genw(M) - 6 passing through go.

Therefore, i1p(h) - p(h) free. 6 D~ takes horizontal vectors to horizontal vectors and vertical vectors to vertical vectors. PROOF: A horizontal vector on TgM_ 1 or TJA means a transverse traceless vector, hence the first part of the theorem. 5. The Principal Bundles of Teichmiiller Theory Differentiation with respect to t at t = 0 yields • Surprisingly, we can put another Riemannian structure on M~l which makes -q, an isometry. Unlike the L2-metric, it is non-degenerate on M~l' but not on MO. It is defined by (((h, k}}}g = JtT(HT KT)d/Lg for h, k E TgM~l M where HT = H - (ltT H) .

This implies that tic differential. pT = Re ((u + iv) . (dx + idyn = Re~(z)dz2 is a holomorphic quadra- We therefore clearly have a bijective correspondence pT +-> ~(z )dz2 between SiT(go) and the space of holomorphic quadratic differentials on (M,c(go)). Moreover, since holomorphic implies Coo it follows that SfT(gO) consists of only Coo tensors (cf. 2 above). By the Riemann-Roch theorem, the space of holomorphic quadratic differentials has dimension 6 genus(M) - 6. This fact will be considered in more detail in the appendix.