## The perturbation of and metric and standard bubble

Suppose , then it is well know that it satisfies

on . Now we want to perturb the Euclidean metric a little bit and an approximate solution of above equation. Suppose where is a symmetric 2-tensor and is the Euclidean metric. The reasonable equation of must satisfy should be

Suppose . Taking derivative on both sides of the above equation and restrict to , one can get the equation must satisfy

where we have use the fact that

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