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Terry Tao
Terry Tao
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One example I know of is the KazhdanKazdan-Warner identity: if $(M,g)$ is a Riemannian surface, with scalar curvature $R$, and $X$ is a conformal Killing vector field, then $\int_M R \operatorname{div}(X)\ dg = 0$. You would think that this identity could just be proven by clever integration by parts, but as far as I know there is no such "local" proof known of this identity. The only proof I know of is to invoke the uniformisation theorem to show that $(M,g)$ is conformal to a constant curvature manifold (for which the identity is just an easy integration by parts). One then deforms the original manifold conformally to the constant curvature manifold and verifies that $\int_M R \operatorname{div}(X)\ dg$ is constant with respect to this perturbation.

(The same method can be used to prove more basic global identities of this type, such as the Gauss-Bonnet theorem, although for such identities, intrinsic proofs not requiring deformation are certainly available.)

Among other things, the KazhdanKazdan-Warner identity can be used to help classify asymptotic shrinking Ricci solitons in two dimensions, which ends up being one of the components of Perelman's proof of the Poincare conjecture (see my notes on this topic).

One example I know of is the Kazhdan-Warner identity: if $(M,g)$ is a Riemannian surface, with scalar curvature $R$, and $X$ is a conformal Killing vector field, then $\int_M R \operatorname{div}(X)\ dg = 0$. You would think that this identity could just be proven by clever integration by parts, but as far as I know there is no such "local" proof known of this identity. The only proof I know of is to invoke the uniformisation theorem to show that $(M,g)$ is conformal to a constant curvature manifold (for which the identity is just an easy integration by parts). One then deforms the original manifold conformally to the constant curvature manifold and verifies that $\int_M R \operatorname{div}(X)\ dg$ is constant with respect to this perturbation.

(The same method can be used to prove more basic global identities of this type, such as the Gauss-Bonnet theorem, although for such identities, intrinsic proofs not requiring deformation are certainly available.)

Among other things, the Kazhdan-Warner identity can be used to help classify asymptotic shrinking Ricci solitons in two dimensions, which ends up being one of the components of Perelman's proof of the Poincare conjecture (see my notes on this topic).

One example I know of is the Kazdan-Warner identity: if $(M,g)$ is a Riemannian surface, with scalar curvature $R$, and $X$ is a conformal Killing vector field, then $\int_M R \operatorname{div}(X)\ dg = 0$. You would think that this identity could just be proven by clever integration by parts, but as far as I know there is no such "local" proof known of this identity. The only proof I know of is to invoke the uniformisation theorem to show that $(M,g)$ is conformal to a constant curvature manifold (for which the identity is just an easy integration by parts). One then deforms the original manifold conformally to the constant curvature manifold and verifies that $\int_M R \operatorname{div}(X)\ dg$ is constant with respect to this perturbation.

(The same method can be used to prove more basic global identities of this type, such as the Gauss-Bonnet theorem, although for such identities, intrinsic proofs not requiring deformation are certainly available.)

Among other things, the Kazdan-Warner identity can be used to help classify asymptotic shrinking Ricci solitons in two dimensions, which ends up being one of the components of Perelman's proof of the Poincare conjecture (see my notes on this topic).

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Terry Tao
Terry Tao
  • 114.1k
  • 33
  • 462
  • 539

One example I know of is the Kazhdan-Warner identity: if $(M,g)$ is a Riemannian surface, with scalar curvature $R$, and $X$ is a conformal Killing vector field, then $\int_M R \operatorname{div}(X)\ dg = 0$. You would think that this identity could just be proven by clever integration by parts, but as far as I know there is no such "local" proof known of this identity. The only proof I know of is to invoke the uniformisation theorem to show that $(M,g)$ is conformal to a constant curvature manifold (for which the identity is just an easy integration by parts). One then deforms the original manifold conformally to the constant curvature manifold and verifies that $\int_M R \operatorname{div}(X)\ dg$ is constant with respect to this perturbation.

(The same method can be used to prove more basic global identities of this type, such as the Gauss-Bonnet theorem, although for such identities, intrinsic proofs not requiring deformation are certainly available.)

Among other things, the Kazhdan-Warner identity can be used to help classify asymptotic shrinking Ricci solitons in two dimensions, which ends up being one of the components of Perelman's proof of the Poincare conjecture (see my notes on this topic).