On Area Growth in Sol

Abstract

Let Sol be the $3$-dimensional solvable Lie group whose underlying space is ${\mathbb{R}}^3$ and whose left-invariant Riemannian metric is given by $$e^{-2z}d{x}^2+e^{2z}d{y}^2+d{z}^2.$$ Building on previous joint work with Matei Coiculescu, which characterizes the cut locus in Sol, we prove that the sphere of radius r in sol has area at most $20\pi e^r$ provided that r is sufficiently large. This estimate is sharp up to a factor of 10

The previous version of the paper had a bound of $611e^r$. In this version, I improve the constant from $611$ to $20\pi$. This is an order of magnitude better. The proof is essentially the same. I am just more careful with the estimates in a few places

Cited by in corpus (2)

• An Interpolation from Sol to Hyperbolic Space
• Some New Results in Geometric Analysis