positive mass theorem

E603722

mathematical theorem result in differential geometry result in general relativity

The positive mass theorem is a fundamental result in differential geometry and general relativity stating that, under suitable conditions, the total mass of an isolated gravitational system is nonnegative and vanishes only for flat spacetime.

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All labels observed (6)

Label	Occurrences
ADM mass is nonnegative for suitable asymptotically flat manifolds	1
Lorentzian positive mass theorem	1
Riemannian positive mass theorem	1
positive mass theorem canonical	1
positive mass theorem in general relativity	1
positive mass theorem with charge	1

How this entity was disambiguated

This entity first appeared as the object of triple T6514674 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: positive mass theorem
Context triple: [Richard Schoen, notableWork, positive mass theorem]

A. Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
B. Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
C. Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
D. geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
E. Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: positive mass theorem
Target entity description: The positive mass theorem is a fundamental result in differential geometry and general relativity stating that, under suitable conditions, the total mass of an isolated gravitational system is nonnegative and vanishes only for flat spacetime.

A. Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
B. Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
C. Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
D. geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
E. Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in differential geometry ⓘ result in general relativity ⓘ
alsoKnownAs	positive energy theorem ⓘ
alternativeProofMethod	spinor methods ⓘ
appliesTo	initial data sets for Einstein field equations ⓘ time-symmetric initial data in the Riemannian case ⓘ
asserts	total mass is nonnegative under suitable conditions ⓘ total mass vanishes only for flat spacetime ⓘ
assumes	asymptotically flat spacetime ⓘ dominant energy condition ⓘ
concerns	ADM mass ⓘ Bondi mass ⓘ total mass of an isolated gravitational system ⓘ
ensures	ADM mass is nonnegative for suitable asymptotically flat manifolds ⓘ linked to: positive mass theorem
field	differential geometry ⓘ general relativity ⓘ mathematical relativity ⓘ
firstCompleteProofYear	1979 ⓘ
generalizedBy	positive mass theorem in higher dimensions ⓘ positive mass theorem with charge ⓘ linked to: positive mass theorem
hasVersion	Lorentzian positive mass theorem ⓘ linked to: positive mass theorem Riemannian positive mass theorem ⓘ linked to: positive mass theorem minimal surface proof version ⓘ spinorial proof version ⓘ
implies	Minkowski spacetime is the unique zero-mass solution under the hypotheses ⓘ linked to: Minkowski space-time no negative-mass isolated gravitational systems under the hypotheses ⓘ stability of Minkowski spacetime with respect to mass ⓘ
influenced	geometric analysis ⓘ global differential geometry ⓘ mathematical foundations of general relativity ⓘ
originalProofMethod	minimal surface techniques ⓘ
provedBy	Edward Witten ⓘ Richard Schoen ⓘ Shing-Tung Yau ⓘ
relatedTo	Penrose inequality ⓘ Yamabe problem ⓘ cosmic censorship conjecture ⓘ
requiresCondition	appropriate decay of the metric at infinity ⓘ nonnegative scalar curvature in the Riemannian version ⓘ
spinorialProofYear	1981 ⓘ
usesConcept	ADM 4-momentum ⓘ asymptotically flat Riemannian manifold ⓘ energy conditions ⓘ scalar curvature ⓘ
zeroMassCase	manifold is isometric to Euclidean space in the Riemannian version ⓘ spacetime is isometric to Minkowski spacetime in the Lorentzian version ⓘ

How these facts were elicited