The Isoperimetric Problem on Planes With Density

Abstract:Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space R 3 with a positive density function e φ , where φ = −x 2 − y 2 , (x, y, z) ∈ R 3 and construct all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted Gaussian curvature and the mean curvature functions. As a result, we give a classification of weighted minimal helicoidal surfaces as well as examples of helicoidal surfaces with some weighted Gaussian curvature and mean curvature functions in the space.

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“…It follows that a helicoidal surface is a helicoid. Otherwise, we can easily compute Equation (11), in such case f is given by (12).…”

Section: Corollarymentioning

confidence: 99%

“…For more details about manifolds with density and some relative topics, we refer readers to [12][13][14][15][16][17], etc. In particular, Hieu and Hoang [13] studied ruled surfaces and translation surfaces in R 3 with density e z and they classified the weighted minimal ruled surfaces and translation surfaces.…”

Section: Introductionmentioning

confidence: 99%

Constructions of Helicoidal Surfaces in Euclidean Space with Density

Yoon

Kim

et al. 2017

Symmetry

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“…In [2], Theorem 8 was proved by using the property "in a plane with nonconstant density e ϕ such that G ϕ = 0, an isoperimetric region has no compact components". We shall give an alternative proof for this result.…”

Section: Theorem 8 ([2]) the Plane With Density Ementioning

confidence: 99%

Some results on curves in the plane with log-linear density

Nam

2017

Asian-European J. Math.

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In this paper, we show some relations between traveling fronts of curvature flow and constant ϕ-curvature curves. In the plane with density e ϕ satisfying ∆ϕ = 0, we prove a Fenchel's type theorem for the class of simple, closed, convex curves. We also prove that among all curves joining two given points, a curve minimizes weighted arc-length if and only if its ϕ-curvature is 0. Another proof is also presented for the fact "the plane with density e x contains no isoperimetric region".

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“…A surface with H D 0 is called a weighted minimal surface or a -minimal surface in E 3 . For more details about manifolds with density and some relative topics we refer to [6][7][8][9][10][11][12]. In particular, Hieu and Hoang [8] studied ruled surfaces and translation surfaces in E 3 with density e z and they classified the weighted minimal ruled surfaces and the weighted minimal translation surfaces.…”

Section: Introductionmentioning

confidence: 99%