Pointwise energy monotonicity for stable Higgs bundles

Abstract

We study the Dai--Li conjecture concerning the pointwise monotonicity of the energy density along the ${\mathbb{C}}^{\ast }$-flow of stable $\mathrm{SL}(n,\mathbb{C})$ Higgs bundles. We prove that the conjecture holds in rank two: for every stable $\mathrm{SL}(2,\mathbb{C})$ Higgs bundle, the energy density is pointwise nondecreasing along the ${\mathbb{C}}^{\ast }$-orbit. In contrast, we show that this phenomenon is genuinely rank-dependent. For every rank $n\ge 3$, we construct stable $\mathrm{SL}(n,\mathbb{C})$ Higgs bundles for which the energy density fails to be monotone along the ${\mathbb{C}}^{\ast }$-flow.

arXiv comment12 pages, comments are welcome