The object
In the usual holographic prescription, an extremal surface measures an area. A defect extremal surface adds a second contribution: the entropy of quantum degrees of freedom localized on the defect. The preferred surface is the one that extremizes the combined quantity.
The geometry and the defect theory therefore cannot be optimized separately. Moving the surface changes where it meets the defect, which changes the quantum term, which in turn shifts the preferred geometry.
Why the defect matters
A defect is more than a marked line. It carries degrees of freedom and changes the variational problem. This is why an RT surface that crosses or ends on a defect requires a corrected prescription.
In the published AdS/BCFT construction, this combined extremization reproduces the boundary quantum extremal surface result. The match gives the island formula a concrete holographic counterpart in that setup.
Suggested citation
Deng, Feiyu. “What a defect extremal surface is.” feiyudeng.com, 2026-07-13.