A relative isoperimetric inequality for certain warped product spaces

Shawn Rafalski 1
  • 1 Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT 06824, USA

Abstract

Given a warped product space ℝ ×f N with logarithmically convex warping function f, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a k-dimensional region F ⊂ {b} × N , and the horizontal graph C ⊂ ℝ ×f N of an almost everywhere differentiable map over F, we prove that the k-volume of C is always at least the k-volume of the smooth constant height graph over F that traps the same (1 + k)-volume above F as C. We use this to solve a Dido problem for graphs over vertical fibers, and show that, if the warping function is unbounded on the set of horizontal values above a vertical fiber, the volume trapped above that fiber by a graph C is no greater than the k-volume of C times a constant that depends only on the warping function.

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