The classification of constant weighted curvature curves in the plane with a log-linear density

Abstract

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in R 2 . The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density e y .

Helicoidal Surfaces in Galilean Space With Density

“…Mean and Gaussian curvature for surfaces are one of the main objects, which have geometers interest for along time. A manifold with density is a Riemannian manifold M n with a positive function e φ , known as density, used to weight volume and hypersurface area [2,9]. A nice example of manifolds with density is Gauss space, the Euclidean space with Gaussian probability density…”

“…where H is the Riemannian mean curvature of the hypersurface [9]. The weighted mean curvature H φ of a surface in E 3 with density e φ was introduced by Gromov [10], and it is a natural generalization of the mean curvature H of a surface.…”

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

“…The classification of constant weighted curvature curves in a plane with a log-linear density has been done in [7] and some other results, such as Fenchel's type theorem for the class of simple, closed, convex curves and the fact "the plane with density e x contains no isoperimetric region" have been proved in [14]. 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.…”