An elementary proof of MinVol(ℝn) = 0 for n ≥ 3

Abstract

In this paper, we give an elementary proof of the result that the minimal volumes of ℝ3 and ℝ4 are zero. The approach is to construct a sequence of explicit complete metrics on them such that the sectional curvatures are bounded in absolute value by 1 and the volumes tend to zero. As a direct consequence, we get that MinVol (ℝn) = 0 for n ≥ 3.