Example 10.25.1136258B2EA152

Example 10.25.11.36258B2EA152 Fix $\lambda >0$ and $0<A<1/\lambda $, and take $u_{\mathrm c}=\pi _1^*v_{\mathrm c}$ with the background of lemma 10.25.9. This gives a complete metric on $M$ solving

where the right side is extended by zero on $D$. The three exponents in (10.25.6) can be taken to be

so their sum equals the critical exponent $2$.

Indeed, define $f_{\mathrm c}=\mathrm{e}^{-\lambda u_{\mathrm c}}\det _\theta (\theta +\mathrm{dd}^{\mathrm{c}}u_{\mathrm c})$ on $M$. On the inner disk the product structure gives

The substitution $L=-\log r^2$ in the radial integral shows that the second term is integrable precisely when $\lambda A<1$. The first term is integrable as well, and the density is smooth elsewhere. The identity of measures across $D$ follows from the preceding lemma. Since $\tau =L+K$, all three bounds hold with the stated exponents, and completeness follows from corollary 10.25.5. Along a radial normal segment the divergence of the length at the critical exponent is also visible directly:

For arbitrary paths, however, completeness requires the corollary, not just this family of radial paths.