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Abstract. Existence of global CMC foliations of constant curvature 3-dimensional maximal globally hyperbolic Lorentzian manifolds, containing a constant mean curvature hypersurface with genus(Σ) > 1 is proved. Constant curvature 3-dimensional Lorentzian manifolds can be viewed as solutions to the 2+1 vacuum Einstein equations with a cosmological constant. The proof is based on the reduction of the corresponding Hamiltonian system in constant mean curvature gauge to a time dependent Hamiltonian system on the cotangent bundle of Teichmüller space. Estimates of the Dirichlet energy of the induced metric play an essential role in the proof.
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PrefaceThis book is the second volume of a treatise on minimal surfaces consisting of altogether three volumes which can be read and studied independently of each other. The central theme is boundary value problems for minimal surfaces such as Plateau's problem. The present treatise forms a greatly extended version of the monograph Minimal Surfaces I, II by U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, published in 1992, which is often cited in the literature as [DHKW]. New coauthors are Friedrich Sauvigny for the first volume and Anthony J. Tromba for the second and third volume.The four main topics of this second volume are free boundary value problems, regularity of minimal surfaces and their geometric properties, and finally a new method is introduced to show that minimizers of area are immersed. Since minimal surfaces in R 3 are understood as harmonic, conformally parametrized mappings X : Ω → R3 of an open domain Ω in R 2 , they are real analytic in Ω, and so the problem of smoothness for X is the question how smooth X is at the boundary ∂Ω if X is subject to certain boundary conditions. However, even if X is "analytically regular", it might not be "geometrically regular" since it could have branch points. We investigate how X behaves in the neighbourhood of branch points, and secondly whether such points actually exist. In addition we describe geometric properties of minimal surfaces in R 3 or, more generally, of H-surfaces in an n-dimensional Riemannian manifold. This book can be read independently from the preceding volume of this treatise although we use some terminology and results from the previous material.We thank E. Kuwert, F. Müller, D. Schwab, H. von der Mosel, D. Wienholtz, and S. Winklmann for pointing out errors and misprints in [DHKW] which are corrected here. Particularly we are indebted to Frank Müller for some penetrating contributions to Chapter 3, and to Albrecht Küster who...
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