Teichmüller Theory in Riemannian Geometry

Abstract

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A construction of the Deligne-Mumford orbifold

“…The resulting functions ξ and η are continuous across the circle of of radius r by (18). Hence they are holomorphic on the large annulus r 2 < |z| < 1.…”

“…Let B be an open neighborhood of 0 in C 3g−3 and ι : B → J be a local holomorphic slice such that ι(0) = j 0 (see [18] or Section 9). Then the projection Q ι → B is a local coordinate chart on Q → T .…”

“…This tangent space is clearly a complex vector space (the complex structure isĵ → jĵ) and it is not hard to show (see e.g. [18] or Section 9) that this almost complex structure on J (Σ) is integrable and that the action admits a holomorphic 4 slice through every point. Since the action of Diff 0 (Σ) is (tautologically) by holomorphic diffeomorphisms of J (Σ), this defines a complex structure on the base T g which is independent of the choice of the local slice used to define it.…”

“…In [2] Earle showed that there is no global holomorphic section of J (Σ) → T g . The monograph of Tromba [18] contains a nice exposition of this point of view (and more) and the anthology [4] is very helpful for understanding the history of the subject and other points of view. Now we take a different point of view.…”

Holomorphic Riemannian Maps

“…One is Calabi-Yau manifolds which have their applications in superstring theory [3]. The other one is Teichmuler spaces applicable to relativity [14]. It is also important to note that CR-structures have been extensively used in spacetime geometry of relativity [12].…”