On the boundaries of special Lagrangian submanifolds

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dc.contributor.advisorHarvey, F. Reeseen_US
dc.creatorFu, Leien_US
dc.date.accessioned2009-06-04T00:46:43Zen_US
dc.date.available2009-06-04T00:46:43Zen_US
dc.date.issued1995en_US
dc.description.abstractAn n-dimensional submanifold M in ${\bf C}\sp{n} = {\bf R}\sp{2n}$ is called Lagrangian if the restriction of $\omega$ to M is zero, where $\omega = \Sigma{\limits\sb{i}}dz\sb{i}\ \wedge\ d\bar z\sb{i}$. It is called special Lagrangian if the restrictions of $\omega$ and Imdz = Im($dz\sb1 \wedge \cdots \wedge dz\sb{n}$) are zero. Special Lagrangian submanifolds are volume minimizing, and conversely, any minimal Lagrangian submanifold can be transformed to a special Lagrangian submanifold by a unitary transformation. This paper studies the conditions satisfied by the boundaries of special Lagrangian submanifolds. We say an $n-1$ dimensional submanifold N in ${\bf C}\sp{n}$ satisfies the moment condition if $\int\sb{N}\ \phi$ = 0 for any $n-1$ form $\phi$ with $d\phi$ belonging to the differential ideal generated by $\omega$ and Imdz. By Stokes' formula, the boundary of a special Lagrangian submanifold satisfies the moment condition. In fact on ${\bf C}\sp2,$ the converse is also true, that is, a curve is the boundary of a special Lagrangian submanifold if it satisfies the moment condition. However, we show in this paper that for $n\ge3,$ there exist $n-1$ dimensional submanifolds which satisfy the moment condition but do not bound any special Lagrangian submanifolds. Let f be a real valued function defined on some open subset on ${\bf C}\sp{n}$ which contains a special Lagrangian submanifold M. We show that $$\Delta\sb{M}f = d(J(\nabla f)\quad\rfloor\ {\rm Im}dz){\vert}\sb{M},$$where $\Delta\sb{M}$ is the Laplace operator on M and J is the standard complex structure. Using this formula and the maximum principle, we prove that if $d(J(\nabla f)\quad\rfloor$ Imdz) belongs to the differential ideal generated by df, $\omega$ and Imdz, then $f{\vert}\sb{M}$ attains its maximum and minimum on the boundary of M. This result is then used to produce some conditions other than the moment condition which are satisfied by the boundaries of special Lagrangian submanifolds.en_US
dc.format.extent37 p.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.callnoTHESIS MATH. 1995 FUen_US
dc.identifier.citationFu, Lei. "On the boundaries of special Lagrangian submanifolds." (1995) Diss., Rice University. <a href="https://hdl.handle.net/1911/16823">https://hdl.handle.net/1911/16823</a>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/16823en_US
dc.language.isoengen_US
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.en_US
dc.subjectMathematicsen_US
dc.titleOn the boundaries of special Lagrangian submanifoldsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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