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Alexandre Eremenko
Alexandre Eremenko
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The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is called the Nierenberg problem, and the complete answer is not known.

For the conditions of solvability of your equation in this case, see

Kazdan, Jerry L.; Warner, F. W. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.

For recent surveys, see https://arxiv.org/pdf/1411.5743.pdf and https://arxiv.org/pdf/1707.02938.pdf

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic.

For the conditions of solvability of your equation in this case, see

Kazdan, Jerry L.; Warner, F. W. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is called the Nierenberg problem, and the complete answer is not known.

For the conditions of solvability of your equation in this case, see

Kazdan, Jerry L.; Warner, F. W. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.

For recent surveys, see https://arxiv.org/pdf/1411.5743.pdf and https://arxiv.org/pdf/1707.02938.pdf

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.

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Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is indeed true, and follows from

For the resultsconditions of Troyanovsolvability of your equation in this case, Prescribing curvature on compact surfaces with conical singularities see

Kazdan, Jerry L.; Warner, F. W. TransCurvature functions for compact 2-manifolds. Amer Ann. of Math. Soc. 324 (19912), no. 2 99 (1974), 793–82114–47.

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is indeed true, and follows from the results of Troyanov, Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821.

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic.

For the conditions of solvability of your equation in this case, see

Kazdan, Jerry L.; Warner, F. W. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.

Source Link
Alexandre Eremenko
Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429

The theorem you stated can be true only for genus zero (that is for the sphere), if $K(x)<0$ at some point $x$); this follows from the Gauss Bonnet theorem that integral of the curvature $-K$ is equal to the Euler characteristic. It is indeed true, and follows from the results of Troyanov, Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821.

To obtain such metrics on surfaces of higher genus, one has to permit conic singularities of the metric. This is a hot research topic nowadays, and there is a recent survey.