What is the boundary around a black hole that nothing can escape?

The boundary around a black hole that nothing can escape is the event horizon.

In general relativity, the event horizon is a boundary in spacetime, not a solid surface. Once an object or light ray crosses it, every possible future-directed path leads farther inward, so no signal can reach a distant observer. The popular description that the escape velocity exceeds the speed of light is useful as an analogy, but the deeper explanation involves the way gravity curves spacetime and tips light cones inward.

For a non-rotating black hole, the event horizon lies at the Schwarzschild radius, whose size depends directly on the black hole’s mass. A distant observer may see an approaching object appear to slow and fade because of gravitational redshift, while the falling object crosses the horizon locally without encountering a physical wall.

The alternatives describe different ideas. The ergosphere surrounds a rotating black hole and permits energy extraction; the photon sphere is a region where light can orbit; and the singularity is the central region predicted by the idealized classical solution. None is the escape boundary itself.