The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

Abstract

Abstract. We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

Asymptotic dimension of planes and planar graphs

“…In this paper we deal with asymptotic dimension in a purely geometric setting, that of Riemannian planes and planar graphs. An aspect of the geometry of Riemannian planes that is studied extensively is that of the isoperimetric problem-even though in that case one usually imposes some curvature conditions (see [3], [15], [12], [18], [10], [9]). We note that Bavard-Pansu ( [2], see also [4]) have calculated the minimal volume of a Riemannian plane.…”

Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere

“…On the other hand, for the sharp interface version E 0 it was shown that the global minimizer of the classical isoperimetric problem on S 2 is the single cap, i.e., the set with boundary consisting of a single circle (cf. [17,26]). Also recently, in [4], the authors established the stability of the isoperimetric domains on S 2 by proving a quantitative version of the isoperimetric inequality on the sphere.…”

The isoperimetric problem in complete annuli of revolution with increasing Gauss curvature

“…This result was recovered by different methods by Pansu [9], Topping [12], Morgan et al . [8] and Ritoré [10].…”

“…Even in this simple class of examples, the geometry of the perimeter-minimizing regions of given area can be quite complex. In some planes [8] and spheres [10] of revolution, these regions can be either discs or annuli, and in annuli of revolution with decreasing curvature from one end of finite area the isoperimetric regions are bounded by a single circle of revolution [8,10]. On the other hand, in tori of revolution, the boundary of a stable region can be composed of curves of constant geodesic curvature which are not circles of revolution [4].…”

“…The case of spheres follows from the classical isoperimetric inequality. Ellipsoids were treated in [10, § 3], and also partially in [8]. Finally, for the one-sheeted hyperboloids, the solutions are described in theorem 4.1.…”