The VMO-Teichm\"uller space and the variant of Beurling-Ahlfors extension by heat kernel

We give a real-analytic section for the Teichm"uller projection onto the VMO-Teichm"uller space by using the variant of Beurling-Ahlfors extension by heat kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this result, we prove that the VMO-Teichm"uller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichm"uller space admits a real-analytic contraction mapping.