### Free boundary stable hypersurfaces in manifolds with density and rigidity

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### Isoperimetric problems in sectors with density r^p

, 2010

"... We consider the isoperimetric problem in the plane with density r p, p> 0, and prove that the solution is a circle through the origin. We use the stability of this isoperimetric curve to prove an ..."

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We consider the isoperimetric problem in the plane with density r p, p> 0, and prove that the solution is a circle through the origin. We use the stability of this isoperimetric curve to prove an

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

, 2013

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### ON ISOPERIMETRIC INEQUALITIES FOR LOG-CONVEX MEASURES

, 805

"... Abstract. We study isoperimetric inequalities for measures of the type µ = e V dx, where V is convex. Using diverse techniques we estimate isoperimetric profiles for a broad class of such measures. The main result is obtained by the optimal transportation method. ..."

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Abstract. We study isoperimetric inequalities for measures of the type µ = e V dx, where V is convex. Using diverse techniques we estimate isoperimetric profiles for a broad class of such measures. The main result is obtained by the optimal transportation method.

### Table of Contents

"... This manual is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This manual is distributed in the hope that it will be useful ..."

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This manual is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This manual is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this manual if not, write to the Free Software

### Isoperimetric Regions in Gauss Sectors

, 2007

"... We consider the free boundary isoperimetric problem in sectors of the Gauss plane. The solution is not always a circular arc as in sectors of the Euclidean plane. We prove that the solution is sometimes a ray and we conjecture that the solution is sometimes a ”rounded n-gon ” which we discovered com ..."

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We consider the free boundary isoperimetric problem in sectors of the Gauss plane. The solution is not always a circular arc as in sectors of the Euclidean plane. We prove that the solution is sometimes a ray and we conjecture that the solution is sometimes a ”rounded n-gon ” which we discovered computationally using Mathematica. 1

### On the Four Vertex Theorem on planes with radial density e ϕ(r)

, 2008

"... It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every r ..."

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It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every radial density. A manifold with density is a Riemannian manifold M n with a positive density function e ϕ(x) used to weight volume and hypersurface area. Such manifolds appeared in many ways in mathematics, for example as quotients of Riemannian manifolds or as Gauss space. Gauss space Gn n x2 is Euclidean space with Gaussian probability density (2π) 2 e 2 that is very interesting to probabilists. For more details about manifolds with density and some first results in Morgan’s grand goal to “generalize all of Riemannian geometry to manifolds with density ” we refer the reader to [6], [7], [4], [1]. Following Gromov ([5]) the natural generalization of the mean curvature of hypersufaces on a manifold with density eϕ is defined as Hϕ = H − 1 dϕ (1) n − 1 dn and therefore, the curvature of a curve on a plane with density eϕ is kϕ = k − dϕ. (2) dn We call kϕ the curvature with density or ϕ−curvarure of the curve. In this note, we study the Four Vertex Theorem on planes with radial density e ϕ(r), where r is the distance from the origin. Curves and the function ϕ are assumed to be of the class C 3 and C 2, respectively. It is well known that “every simple closed curve on Euclidean plane has at least four vertices ” (the Four Vetex Theorem). This theorem has a long and interesting history (see [3], [8]). 1 Figure 1: A circle has 2 vertices. First we observed that in general, the Four Vertex Theorem does not hold on planes with density. On the Gauss plane G 2 with density e ϕ = 1 2π e−r2 /2, let α: [0, 2π] − → R 2 t ↦− → (R cos(t), R sin(t) + b) be a parametrization of the circle with center I(0, b) and radius R. On the Gauss plane G 2, direct computation shows that Applying 3, we get the ϕ-curvature of the circle kϕ = x ′ y ′ ′ − x ′ ′ y ′ + xy ′ − x ′ y. (3) kϕ = 2R + R sin(t). The equation k ′ ϕ = 0 has exactly two solutions π