Surfaces in R3 with density

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 .…”

Non-Null Curves With Constant Weighted Curvature in Lorentz-Minkowski Plane With Density

“…In [10], Lopez has studied the minimal surfaces in Euclidean 3-space with a log-linear density φ(x, y, z) = αx + βy + γz, where α, β and γ are real numbers not all-zero. Also, Belarbi et al have studied the surfaces in R 3 with density and they have given some results in a Riemannian manifold M with density in [1] and [2], respectively. Furthermore, ruled and translation minimal surfaces in R 3 with density e z ; helicoidal surfaces in R 3 with density e −x 2 −y 2 and weighted minimal affine translation surfaces in Euclidean space with density have been studied in [6,18,19], respectively.…”

Rotational Surfaces Generated by Planar Curves in E3 with Density

“…where G is Gaussian curvature of the surface and ∆ is the Laplacian operator. For another characterizations of manifolds with density, we refer to [3], [9], [10], [11], [13], [16], [20] and etc.…”