Mikhail Gromov's filling area conjecture, in differential geometry, asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementAmong orientable Riemannian surfaces that isometrically fill a boundary curve of a given length, the hemisphere has the minimum area. 1 Proposed Year Progress Toward ResolutionGromov proved in the same paper that the hemisphere has least area among Riemannian disks that isometrically fill a circle of given length; the general case remains a conjecture. 2 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Filling Area Conjecture (Wikipedia)
Sources
1. Filling Area Conjecture (Wikipedia)
Formal statement section
The hemisphere has minimum area among the orientable compact Riemannian surfaces that fill isometrically their boundary curve, of given length.
- Lead section
View the Source2. Filling Area Conjecture (Wikipedia)
Wikimedia FoundationLead section
Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.
Definitions and statement of the conjecture
Conjecture (Gromov's filling area conjecture, 1983)
Gromov's proof for the case of Riemannian disks
the hemisphere has least area among the Riemannian surfaces that isometrically fill a circle of given length, and are homeomorphic to a disk.
In Branch: Differential Geometry, Lead sentence
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientab
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