Classification of Punctures on Complete Flat Surfaces

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International Electronic Journal of Geometry
Volume:14 Issue:2
Classification of Punctures on Complete Flat Surfaces

Classification of Punctures on Complete Flat Surfaces

Pages : 266-276

View : 39 | Download : 13

Publication Date : 2021-10-29

Article Type : Research Paper

Abstract :We investigate the behavior of a complete flat metric on a surface near a puncture. We call a puncture on a flat surface regular if it has a neighborhood which is isometric to that of a point at infinity of a cone. We prove that there are punctures which are not regular if and only if the curvature at the puncture is $4\pi$. We classify irregular punctures of a flat surface up to modification equivalence, where two punctures are called modification-equivalent if they have isometric neighborhoods. We show that there are uncountably many modification-equivalence classes of punctures on flat surfaces.
Keywords : Flat surface, regular puncture, irregular puncture, conical singularities

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