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.