加边Riemann surfaces的加细Teichmueller空间上的Hilbert流形结构
A Hilbert manifold structure on the refined Teichmueller space of bordered Riemann surfaces
摘要：
我们考虑一类与亏格为g的紧致Riemann曲面双全纯的有边Riemann曲面，其中n个区域双全纯于去掉的圆盘。我们定义了这样的Riemann曲面的一个加细Teichmueller空间，并证明了在2g+2-n>0的情况下，这个加细Teichmueller空间是一个Hilbert流形.从加细Teichmueller空间到通常Teichmueller空间（通常是一个巴拿赫流形）的包含映射是全纯的。我们还证明了共形域论中出现的具有非重叠全纯映射的Riemann曲面的操纵模空间是复Hilbert流形。这个结果需要一个解析的重新制定的模空间，通过扩大非重叠映射的一类映射之间的解析可拓映射和拟共形可拓映射。最后证明了操纵模空间是加细Teichmueller空间与双全纯的真不连续群的商。
英文摘要：
We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmueller space of such Riemann surfaces and demonstrate that in the case that 2g+2-n>0, this refined Teichmueller space is a Hilbert manifold. The inclusion map from the refined Teichmueller space into the usual Teichmueller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally we show that the rigged moduli space is the quotient of the refined Teichmueller space by a properly discontinuous group of biholomorphisms.
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