It is known that every Riemannian metric on a closed manifold is conformal to a metric whose exponential map preserves the Euclidean volume near a point. This thesis concerns the classification problem of such “conformal normal metrics” on a conformal manifold (X, [g]) of dimension n ≥ 3. We first prove the uniqueness of a conformal normal metric within a fixed 1-jet class of metrics. For the proof, we mainly follow Cao’s method in [Cao91] by analyzing a non-linear singular elliptic equation in the framework of weighted Hölder spaces. Our second result concerns the smooth dependence of conformal normal metrics on parameters. As applications, we first construct a smooth Riemannian metric on X × X that is conformal normal near the diagonal on each fiber, and then use this metric to give a simplified proof of the regularity of Habermann’s canonical metric.
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