Return to Answer

1. That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see a historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

For some metrics there are no closed curves of constant geodesic curvatures, ibid. So the optimal disk does not always exist. (I don't know if there are examples with curvature everywhere negative.) On the other hand, on a surface $dr^2 + f(r) d\phi^2$ with negative curvature tending to 0 as $r$ tends to infinity there is no disk of maximum area with a given perimeter.

Also, the domain of maximum area with a given perimeter (if exists) might be non-simply connected, but every component of the boundary must be geodesic anyway...

2. Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

It was later generalized by Alexandrov to the $\mathrm{CAT}(0)$-surfaces, see Section 2.2 of "Geometric inequalities" by Burago and Zalgaller.

3. Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.

1. That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see a historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

For some metrics there are no closed curves of constant geodesic curvatures, ibid. So the optimal disk does not always exist. (I don't know if there are examples with curvature everywhere negative.) On the other hand, on a surface $dr^2 + f(r) d\phi^2$ with negative curvature tending to 0 as $r$ tends to infinity there is no disk of maximum area with a given perimeter.

Also, the domain of maximum area with a given perimeter (if exists) might be non-simply connected, but every component of the boundary must have constant geodesic curvature anyway...

2. Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

It was later generalized by Alexandrov to the $\mathrm{CAT}(0)$-surfaces, see Section 2.2 of "Geometric inequalities" by Burago and Zalgaller.

3. Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.

1. That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see a historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

For some metrics there are no closed curves of constant geodesic curvatures, ibid. So the optimal disk does not always exist. (I don't know if there are examples with curvature everywhere negative.)

On the other hand, on a surface $dr^2 + f(r) d\phi^2$ with negative curvature tending to 0 as $r$ tends to infinity there is no disk of maximum area with a given perimeter.

Also, the domain of maximum area with a given perimeter (if exists) might be non-simply connected, but every component of the boundary must be geodesic anyway.

2. Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

3. Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.

1. That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see a historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

For some metrics there are no closed curves of constant geodesic curvatures, ibid. So the optimal disk does not always exist. (I don't know if there are examples with curvature everywhere negative.) On the other hand, on a surface $dr^2 + f(r) d\phi^2$ with negative curvature tending to 0 as $r$ tends to infinity there is no disk of maximum area with a given perimeter.

Also, the domain of maximum area with a given perimeter (if exists) might be non-simply connected, but every component of the boundary must be geodesic anyway...

2. Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

It was later generalized by Alexandrov to the $\mathrm{CAT}(0)$-surfaces, see Section 2.2 of "Geometric inequalities" by Burago and Zalgaller.

3. Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.

1. That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see the historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

But for some metrics there are no closed curves of constant geodesic curvatures, ibid. (I don't know if there are examples with curvature everywhere negative.) So the optimal disk does not always exist.

2. Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

3. Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.

1. That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see a historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

For some metrics there are no closed curves of constant geodesic curvatures, ibid. So the optimal disk does not always exist. (I don't know if there are examples with curvature everywhere negative.)

On the other hand, on a surface $dr^2 + f(r) d\phi^2$ with negative curvature tending to 0 as $r$ tends to infinity there is no disk of maximum area with a given perimeter.

Also, the domain of maximum area with a given perimeter (if exists) might be non-simply connected, but every component of the boundary must be geodesic anyway.

2. Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

3. Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.