Schramm's Hausdorff-measure problem for SLE: an exact gauge function for the chordal SLE trace, 0 < kappa < 8

For fixed $0<\kappa<8$, $d=1+\kappa/8$, the gauge $h(r)=r^d(\log\log(1/r))^{(2-d)/2}$ satisfies almost surely $0<\mathcal H^h(\gamma([s,t]))<\infty$ for all $0<s<t<\infty$, and $\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))<\infty$. Corollary 1.2: Schramm's suggested gauge $r^d\log\log(1/r)$ is not sigma-finite on any such segment, so the exponent he proposed is wrong. The September 25 companion first proved existence of an exact (non-closed-form) gauge. Not addressed: $\kappa\ge8$, whether the two gauges are equivalent, and the relation of $\mathcal H^h$ to natural parametrization or Minkowski content.

For fixed $0<\kappa<8$, $d=1+\kappa/8$, the gauge $h(r)=r^d(\log\log(1/r))^{(2-d)/2}$ satisfies almost surely $0<\mathcal H^h(\gamma([s,t]))<\infty$ for all $0<s<t<\infty$, and $\mathbb E\mathcal H^h(\Gamma\cap\overline B(0,R))<\infty$. Corollary 1.2: Schramm's suggested gauge $r^d\log\log(1/r)$ is not sigma-finite on any such segment, so the exponent he proposed is wrong. The September 25 companion first proved existence of an exact (non-closed-form) gauge. Not addressed: $\kappa\ge8$, whether the two gauges are equivalent, and the relation of $\mathcal H^h$ to natural parametrization or Minkowski content.