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In episode 3 or 4 of Avatar: The Last Airbender, Aang enters the avatar state upon seeing the remains of his old master surrounded by Firebenders, which results in the eyes of the Avatar statues of the Southern Air Temple glowing as well as various other places in the world.

How are these places and the statues linked to the Avatar? (ie. How are they created? Would damaging them damage the Avatar?)

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Flater is correct in his assessment that the statues are representations of the previous avatars and that them lighting up is more thematic rather than functional.

However, I would like to address the specifics of the question: each of these statues are representations of the previous incarnations of the Avatar. The ones shown outside of the Southern Air Temple are located in other temples scattered around the world, there are at-least one of these temples located in each nation.

It's never explicitly explained how these statues are created, but it is logical to assume the friends and followers (the order of the white lotus) of the latest Avatar are likely the ones who construct these statues.

As for the statues being damaged having any effect on the current Avatar, there is evidence to the contrary. In a later episode of season 1 (episode 7 and 8):

Aang travels to the fire nation to commune with Roku, the previous Avatar, at his temple. At the end of the episode Aang, while channeling Roku, awakens the volcano Roku's temple sits upon, forcing everyone to flee as the temple and the statue inside of burn and melt into the flowing lava. The destruction of this temple and its statue has no lasting effect on Aang.

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You're reading more into into than there is.

The avatar state is something that connects the current and past avatars. It's a sort of mind meld. This is established further in The Legend Of Korra (the next avatar after Aang). Avoiding spoilers, it is explicitly proven that the avatar state is a connection to past avatars.

The statues' eyes light up to showcase the point that Aang is "connecting to the old avatars".

There is no established lore that these statues are physically involved in the process and that the current avatar can be affected by destroying or tampering with the statues. The statues are simply representations of past avatars, and therefore used to non-verbally demonstrate to the viewer that Aang is connecting to the past avatars. It's thematic, not functional.

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