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Defect extremal surface is defined by minimizing the Ryu-Takayanagi surfacecorrected by the defect theory, which is useful when the RT surface crosses orterminates on the defect. Based on the decomposition procedure of a AdS bulkwith a defect brane, proposed in arXiv:2012.07612, we derive Page curve in atime dependent set up of AdS$_3$/BCFT$_2$, and find that the result from islandformula agrees with defect extremal surface formula precisely. We then extendthe study to higher dimensions and find that the entropy computed from bulkdefect extremal surface is generally less than that from island formula inboundary low energy effective theory, which implies that the UV completion ofisland formula gives a smaller entropy in higher dimensions.

Page Curve from Defect Extremal Surface and Island in Higher Dimensions