Bubble function can be defined either on , or . For the following notations, will denote suitable constants which may be different from line to line.

- For every and , define

It is well know that . Moreover is isometric to the standard sphere minus one point.

- For any and define

where is the geodesic distance of and on . Actually

- For each , define by

Both the second and third one satisfy

If we make , the third one will be changed to the second one.

To get the second one from the first one, let us deonte be the stereographic projection from point . Then

It should be able to see the third one from hyperbolic translation directly.

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