The Dirichlet problem for Monge-Ampère type equations on Riemannian manifolds was not previously studied
Research gap analysis derived from 3 mathematics papers in our local library.
The gap
The Dirichlet problem for Monge-Ampère type equations on Riemannian manifolds was not previously studied. Prior work only considered the Monge-Ampère equation in Euclidean space.
Evidence profile
Sourced from the stated research gap and future-work section and abstract of the source papers, classified as general, drawn from work published between 2024 and 2026, spanning 3 journals. Those papers have been cited 3 times in total.
Research trend
Established — well-defined area with open sub-problems.
Supporting evidence — 4 representative gaps
- On some connections between Kobayashi geometry and pluripotential theory (2026) · Transactions of the American Mathematical Society · doi
The paper identifies a gap in understanding the relationships between Kobayashi geometry and the Dirichlet problem for the complex Monge-Ampère equation. The authors highlight the need for further research in this area and provide a direction for future studies.
generalstated research gapevidence 5/5Keywords: paper identifies gap understanding relationships between kobayashi geometry - On some connections between Kobayashi geometry and pluripotential theory (2026) · Transactions of the American Mathematical Society · doi
The paper suggests future research directions, including the further study of connections between Kobayashi geometry and the Dirichlet problem for the complex Monge-Ampère equation. The authors propose exploring the properties of domains with nice boundary geometry and their relationship to the complex Monge-Ampère equation.
generalfuture-work sectionevidence 5/5Keywords: paper suggests future research directions including further study - The Dirichlet Problem for Monge-Ampère Type Equations on Riemannian Manifolds (2026) · Results in Mathematics · doi
The Dirichlet problem for Monge-Ampère type equations on Riemannian manifolds was not previously studied. Prior work only considered the Monge-Ampère equation in Euclidean space.
generalstated research gapevidence 5/5Keywords: dirichlet problem monge-amp type equations riemannian manifolds was - Geodesic connectivity and rooftop envelopes in the Cegrell classes (2024) · Mathematische Annalen · cited 3× · doi
We establish that solutions possessing comparable singularities to the complex Monge–Ampère equation are identical, affirmatively addressing a longstanding open question raised by Cegrell.
generalabstractevidence 4/5Keywords: establish solutions possessing comparable singularities complex monge equation identical affirmatively addressing longstanding open question raised
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