Minimal Translation Surfaces in Sol_3 and Nil_3

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Kasbi Gharahasanlu, Tohid | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 44616 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Fanai, Hamid Reza
Abstract:
A surfaceMin the Euclidean space is called a translation surfaceif it is given by the graph z(s,t)=f(s)+g(t), where f and gare smooth functions on some interval of R. These surfaces are called translation surfaces since its parameterization X(s,t)=(s,t,f(s)+g(t) ) can be written as the sum of two curves (translation), namely , X(s,t)=(s,0,f(s) )+(0,t,g(t) )
In this work , Minimal surfaces in Sol_3 and Nil_3have been studied,where Sol_3and Nil_3are two model geometry of the eight geometries of Thurston. We propose a similar problem in Sol_3 and Nil_3 changing the additive + in the Euclidean space by the group operation * of Sol_3 and Nil_3, such that we have X(s,t)=α(s)*β(t), where α and β are curves contained in coordinate planes and * denotes the group operation. In this work translation surfaces in Sol_3 and Nil_3 whose Mean curvature vanishes have been studied
Keywords:
Translation Surfaces ; Minimal Surfaces ; Mean Curvature ; Sol3 Space ; Nil3 Space

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