jats:titleAbstract</jats:title>jats:pWe give a simple set of geometric conditions on curves jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline1.png" />jats:tex-math$𝜂$</jats:tex-math></jats:alternatives></jats:inline-formula>, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline2.png" />jats:tex-math$𝜂~$</jats:tex-math></jats:alternatives></jats:inline-formula> in jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline3.png" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula> from jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline4.png" />jats:tex-math$0$</jats:tex-math></jats:alternatives></jats:inline-formula> to jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline5.png" />jats:tex-math$\infty$</jats:tex-math></jats:alternatives></jats:inline-formula> so that if jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline6.png" />jats:tex-math$𝜑:H\to H$</jats:tex-math></jats:alternatives></jats:inline-formula> is a homeomorphism which is conformal off jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline7.png" />jats:tex-math$𝜂$</jats:tex-math></jats:alternatives></jats:inline-formula> with jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline8.png" />jats:tex-math$𝜑(𝜂)=𝜂~$</jats:tex-math></jats:alternatives></jats:inline-formula> then jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline9.png" />jats:tex-math$𝜑$</jats:tex-math></jats:alternatives></jats:inline-formula> is a conformal automorphism of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline10.png" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm–Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline11.png" />jats:tex-math$𝜂$</jats:tex-math></jats:alternatives></jats:inline-formula> is a non-space-filling jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline12.png" />jats:tex-math$SLE𝜅$</jats:tex-math></jats:alternatives></jats:inline-formula> curve in jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline13.png" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula> from jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline14.png" />jats:tex-math$0$</jats:tex-math></jats:alternatives></jats:inline-formula> to jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline15.png" />jats:tex-math$\infty$</jats:tex-math></jats:alternatives></jats:inline-formula>, and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline16.png" />jats:tex-math$𝜑$</jats:tex-math></jats:alternatives></jats:inline-formula> is a homeomorphism which is conformal on jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline17.png" />jats:tex-math$H\setminus 𝜂$</jats:tex-math></jats:alternatives></jats:inline-formula>, and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline18.png" />jats:tex-math$𝜑(𝜂)$</jats:tex-math></jats:alternatives></jats:inline-formula>, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline19.png" />jats:tex-math$𝜂$</jats:tex-math></jats:alternatives></jats:inline-formula> are equal in distribution, then jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline20.png" />jats:tex-math$𝜑$</jats:tex-math></jats:alternatives></jats:inline-formula> is a conformal automorphism of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline21.png" />jats:tex-math$H$</jats:tex-math></jats:alternatives></jats:inline-formula>. Applying this result for jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline22.png" />jats:tex-math$𝜅=4$</jats:tex-math></jats:alternatives></jats:inline-formula> establishes that the welding operation for critical (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline23.png" />jats:tex-math$𝛾=2$</jats:tex-math></jats:alternatives></jats:inline-formula>) LQG is well defined. Applying it for jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline24.png" />jats:tex-math$𝜅\in (4,8)$</jats:tex-math></jats:alternatives></jats:inline-formula> gives a new proof that the welding of two independent jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline25.png" />jats:tex-math$𝜅/4$</jats:tex-math></jats:alternatives></jats:inline-formula>-stable looptrees of quantum disks to produce an jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline26.png" />jats:tex-math$SLE𝜅$</jats:tex-math></jats:alternatives></jats:inline-formula> on top of an independent jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1474748019000331_inline27.png" />jats:tex-math$4/𝜅$</jats:tex-math></jats:alternatives></jats:inline-formula>-LQG surface is well defined.</jats:p>