Metrics on Hitchin Components from H\"older Distortion

Abstract

We observe Thurston's asymmetric metric on Teichm\"uller space may be expressed in terms of the H\"older regularity of boundary maps. We then associate $2$-dimensional stratified loci in $\mathbb{RP}^{n-1}$ to $\text{PSL}_n(\mathbb{R})$ Hitchin representations with $n > 3$. We prove that measuring the relative H\"older distortion of these loci gives asymmetric metrics on the Hitchin component $\text{Hit}_n(S)$ with complete and geometrically meaningful symmetrizations. These are the first known geometrically significant complete metrics on $\text{Hit}_n(S)$ for $n > 3$.