Type I ⁺ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density

Abstract

In this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ${\Bbb R}_1^3$with a positive density function. We get a result for minimal case. Also, we give examples of a helicoidal surface with weighted mean curvature and Gaussian curvature.

Rotational Hypersurfaces in Euclidean 4-Space with Density

“…Also, some types of surfaces have been studied by geometers in other spaces such as Minkowski 3-space and Galilean 3-space with density. For instance, a helicoidal surface of type + with prescribed weighted mean curvature and Gaussian curvature in Minkowski 3-space and weighted minimal translation surfaces in Minkowski 3-space with density have been constructed in [25] and [30], respectively. In [31], weighted minimal translation surfaces in the Galilean 3-space with log-linear density have been classified and in [19], weighted minimal and weighted flat surfaces of revolution in Galilean 3-space with density 2 + 2 + 2 have been investigated.…”

Siacci's Theorem for Frenet Curves in Minkowski 3-Space

“…where ε B = B, B L = ±1 and ε T = T, T L = ±1, [5]. Some studies which are studied in the Minkowski 3-dimensional space can be found in [6][7][8][9][10][11][12][13][14].…”

Monge Hypersurfaces in Euclidean 4-Space with Density

“…A hypersurface is called weighted flat (or -flat), if its weighted Gaussian curvature vanishes. After these definitions, lots of studies have been done by differential geometers about weighted manifolds, for instance [16][17][18][19][20][21][22][23][24][25]. Let we take ( ) = (ℎ ) × ( ) −1 .…”