dc.contributor.advisor Harvey, F. Reese Fu, Lei 2009-06-04T00:46:43Z 2009-06-04T00:46:43Z 1995 http://hdl.handle.net/1911/16823 An 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. 37 p. application/pdf eng Mathematics On the boundaries of special Lagrangian submanifolds Thesis Text Mathematics Natural Sciences Rice University Doctoral Doctor of Philosophy Fu, Lei. "On the boundaries of special Lagrangian submanifolds." (1995) Diss., Rice University. http://hdl.handle.net/1911/16823.
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