2023 Fiscal Year Final Research Report
Unification of various theories of zero mean curvature surfaces and exploration of geometrical properties according to signature of metrics
| Project/Area Number |
19K14527
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Nihon University (2020-2023)
Nagoya University (2019) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 微分幾何学 / 光的超曲面 / 極小曲面 / 極大曲面 / 時間的極小曲面 |
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| Outline of Final Research Achievements |
The Calabi-Bernstein theorem on maximal hypersurfaces in Minkowski space is generalized to allow lightlike points where the metric of the surface is degenerate.
It is proved that lightlike hypersurfaces in Lorenzian manifolds with the null energy condition that are lightlike complete are only totally geodesic, and the global structure of such hypersurfaces is clarified. As a class of lightlike hypersurfaces with singularities, we consider the class of null wavefronts and clarify a structure theorem for null wavefronts.
It is shown that there is a one-to-one correspondence, called duality, between the solutions of certain boundary value problems for minimal surfaces in three-dimensional Euclidean space and maximal surfaces in three-dimensional Minkowski spacetime. New reflection principles are also discovered for several related boundary value problems.
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| Free Research Field |
微分幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
ユークリッド空間内の極小曲面およびミンコフスキー空間内の光的曲面，極大曲面，時間的極小曲面という様々なクラスの曲面を平均曲率零曲面という見方から統一的に捉え，異なる空間内の曲面に対して類似した現象や原理があることを解き明かしたことが本研究成果の意義である．また，そのような幾何学的な研究を行うために調和関数論が重要な役割を果たすことが明らかになったことで，曲面の微分幾何学，関数論を始めとした他分野への影響も期待される．
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