Throw a quantum object — a book, an encyclopedia, an entire planet — into a black hole. The black hole emits Hawking radiation and eventually evaporates. Where did the information go? Don Page (1993) computed what the entanglement entropy of the radiation must look like if quantum mechanics is preserved. The 2019 island formula derivation showed this curve actually emerges from gravity itself.
why this curve is the most important plot in quantum gravity
Hawking 1974: black holes emit thermal radiation. Hawking 1976: this radiation is, calculation by calculation, in a mixed (thermal) state — meaning information is destroyed when a black hole evaporates. This violates quantum mechanics, which requires evolution to be unitary (information-preserving).
Don Page (1993): but it can't be destroyed. If you assume the full Hawking radiation + remaining black hole is a pure quantum state evolving unitarily, the entanglement entropy of the radiation alone must follow a specific curve — rising linearly at first (just like Hawking computed), peaking at the half-way point, then falling back to zero as the black hole shrinks below the radiation in size. The radiation eventually purifies itself.
The peak time — when the black hole's entropy equals the radiation's entropy — is the Page time. After the Page time, every new Hawking photon carries information out of the black hole on average. Hawking's calculation can't see this because it ignores gravitational backreaction; the Page curve is what unitarity demands.
2019 — the breakthrough: Penington, and independently Almheiri-Engelhardt-Marolf-Maxfield, computed the gravitational entropy with backreaction using the Ryu-Takayanagi formula generalized for time evolution (the "quantum extremal surface" formula). The Hawking calculation gives the rising part; the new dominant saddle gives the falling part. The Page curve emerges from gravity itself. It's the strongest evidence we have that black hole evaporation preserves information, and it changed how the community thinks about the holographic principle.