Minimal Volume Growth For Rotationally Symmetric Metric

For complete Riemannian manifolds with bounded sectional curvature, Gromov defined the geometric invariant of minimal volume, One can prove that the minimal volume is zero for Euclidean space with dimension greater than two.There are a lot of symmetries for Euclidean space with the standard metric, and the volume of geodesic ball grows polynomially for standard metric. Considering a rotationally symmetric Riemann metric, we can find the minimal volume will not col-lapse to 0.In this paper, we will prove the minimal volume is infinite for Euclidean space with rotationally symmetric metric, if the dimension is greater than two. From the sphere of radius R, by controlling the condition of sectional curvature, get a inequality of sphere volume. We estimate the minimal volume growth is at least linear,, and there is an inequality control an extreme case,And finally, we construct an example, explain the equality case.