On Area Growth in Sol

2020/04/22 by Richard Evan Schwartz, Schwartz, Richard Evan
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.2004.10622

The previous version of the paper had a bound of $611 e^r$. In this version, I improve the constant from $611$ to $20 π$. 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

arxiv created 2021/01/13 · arxiv updated 2021/01/14

Abstract

Let Sol be the 3-dimensional solvable Lie group whose underlying space is ℝ³ and whose left-invariant Riemannian metric is given by e^-2z dx² + e^2z dy² + dz². 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 πe^r provided that r is sufficiently large. This estimate is sharp up to a factor of 10

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