Topology

An Introduction to Compactness Results in Symplectic Field by Casim Abbas

By Casim Abbas

This e-book offers an creation to symplectic box thought, a brand new and critical topic that's presently being built. the place to begin of this thought are compactness effects for holomorphic curves proven within the final decade. the writer provides a scientific advent supplying loads of heritage fabric, a lot of that's scattered during the literature. because the content material grew out of lectures given through the writer, the most goal is to supply an access aspect into symplectic box idea for non-specialists and for graduate scholars. Extensions of yes compactness effects, that are believed to be real by means of the experts yet haven't but been released within the literature intimately, stock up the scope of this monograph.

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38 (Local isometries of H are global isometries) Let U ⊂ H + be an open subset and assume that f : U → H + is of class C 1 , preserves orientation and is a local isometry. Then f is a global isometry (and biholomorphic). Hints: (1) Show first that f must be holomorphic and that it must satisfy the condition f ′ (z) = Im(f (z)) , Im(z) formally, we identify S with S × {0} and we let S ′ := S × {1}. Then ι : S → S ′ is given by ι(x, 0) = (x, 1). 3 More 28 1 Riemann Surfaces (2) Composing with an element in Conf(H) we may assume that f ′ (i) = 1, Im f (i) = 1.

Following similar arguments as before, when we discussed case (i), there is φ ∈ D(π) such that the geodesics γ˜1 and φ(γ˜2 ) intersect transversally. Then the situation is again similar to Fig. 18 and the curves α˜1 , φ(α˜ 2 ) intersect as well. Projecting into S this would imply the contradiction α1 ∩ α2 ̸= ∅. The case where ∂S ̸= ∅ then simply follows from taking the double of S. 65. The statement can be refined as follows: The corresponding geodesics γ1 and γ2 agree as point sets if and only if there are nonzero integers k1 , k2 such that α k1 and α k2 are freely homotopic.

Then − Q i dz ∧ d z¯ = 2y 2 k i=1 αi + (2 − k)π. 61 Let S be a surface diffeomorphic to a nondegenerate pair of pants equipped with a hyperbolic metric h so that ∂S consists of closed geodesics. 48. 58. The arcs γij divide S into two compact simply connected hyperbolic surfaces G1 , G2 . The boundary of each of them consists of six geodesic arcs which intersect orthogonally. 56 both G1 , G2 are isometric to hexagons ˜ 2 ⊂ H. 52. If (S, h) is a stable surface with hyperbolic metric of finite area, we would like to understand what S looks like metrically near its punctures.

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