Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

Abstract

C e n t .E u r .J . M a t h . •1 2 ( 9 ) •2 0 1 4 •1 3 4 9 -1 3 6 1 DO I :1 0 . 2 4 7 8 / s 1 1 5 3 3 -0 1 4 -0 4 1 5 -0 Cen t ra l Eu ropean Jou rna l o f Ma the ma t ics H e l i c o i d a ls u r f a c e s i n M i n k ow s k is p a c ew i t hc o n s t a n t m e a nc u r v a t u r ea n dc o n s t a n tG a u s sc u r v a t u r e Resea rch A r t ic le R a f a e lL ó p e z 1 * ,E sm aD em i r 2 1 D e p a r t am e n t od eG e om e t r í ayT o p o l o g í a ,U n i v e r s i d a dd eG r a n a d a ,1 8 0 7 1G r a n a d a ,S p a i n 2 D e p a r tm e n to f M a t h em a t i c s ,F a c u l t yo fS c i e n c ea n dA r t s ,N e v s e h i rU n i v e r s i t y ,5 0 3 0 0N e v s e h i r ,T u r k e y Rece ived 11 Ap r i l 2013 ; accep ted 12 Dece mbe r 2013 Abs t rac t : W ec l a s s i f ya l lh e l i c o i d a ln o n -d e g e n e r a t es u r f a c e s i n M i n k ow s k is p a c e w i t hc o n s t a n t m e a nc u r v a t u r e w h o s e g e n e r a t i n gc u r v e i sa t h eg r a p ho fap o l y n om i a lo raL o r e n t z i a nc i r c l e . I n t h efi r s tc a s e ,w ep r o v e t h a t t h ed e g r e e o f t h ep o l y n om i a l i s0o r1a n d t h a t t h es u r f a c e i s r u l e d .I f t h eg e n e r a t i n gc u r v e i saL o r e n t z i a nc i r c l e ,w ep r o v e t h a t t h eo n l yp o s s i b i l i t y i s t h a t t h eax i s i ss p a c e l i k ea n d t h ec e n t e ro f t h ec i r c l e l i e so n t h ea x i s .

Null-translation surfaces with constant curvatures in Lorentz-Minkowski 3-space

“…On the other hand, surfaces are additionally analyzed due to different constraints, such as that of constant curvatures, Gaussian or mean, e.g. [5,[8][9][10]. Recently, classification of translation surfaces generated by spatial curves in Euclidean space and with constant Gaussian curvature has been completed [3], while in the Lorentz-Minkowski space it is still an open problem.…”

Generalized Helicoidal Surfaces in Euclidean 5-space

“…In surface with a light-like axis with prescribed mean curvature or Gauss curvature given by smooth function in Minkowski 3-space and solved an open problem left in [3]. Also, in [7], the authors classify all helicoidal non-degenerate surfaces in Minkowski 3-space with constant mean curvature whose generating curve is the graph of a polynomial or a Lorentzian circle. Besides, in [1], the authors studied rotational surfaces in higher dimensional Euclidean spaces.…”

Bour's Theorem of Spacelike Surfaces in Minkowski 4--Space

“…Many mathematicians have obtained significant results regarding helicoidal surfaces in different ambient spaces (see [12,5,7,22,10,23,11,24,26,27,1,2,3]). One of the most notable result is a generalization of the relationship between a right helicoid and a catenoid.…”