Browsing by Author "Harvey, F. Reese"

Now showing 1 - 6 of 6
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    Calibrations on semi-Riemannian manifolds
    (1989) Mealy, Jack G.; Harvey, F. Reese
    This thesis "dualizes" Harvey and Lawson's notion of calibrated geometry on a Riemannian manifold to the semi-Riemannian category. By considering the appropriate spaces (with signature) analogous to the positive definite situations, we prove inequalities which in turn lead to analogues of the main examples discussed by the aforementioned. These are: complex geometry on C$\sp{p,q},$ special Lagrangian geometry on R$\sp{n,n}$, associative and coassociative geometries on the imaginary split octonians, and Cayley geometry on the split octonians. By nature of these inequalities, the $\phi$-submanifolds in all of these examples are volume maximizing in an appropriate sense, which contrasts with the minimizing property in the positive definite situation. The PDE's associated with these geometries are derived, and are seen to resemble their positive definite analogues. Examples of $\phi$-submanifolds are subsequently discussed. The contact sets $\{\phi \equiv 1\}\ \cap$ Grassmannian in the positive definite and signature cases are also seen to exhibit a duality in the sense of Riemannian globally symmetric spaces. Indeed, the dual nature of the semi-Riemannian category with the Riemannian category is emphasized throughout. However, this "duality" is not precise. There are important calibrations in the positive definite category whose would-be-duals in the signature cases are not calibrations.
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    Compactification problems in the theory of characteristic currents associated with a singular connection
    (1993) Zweck, John; Harvey, F. Reese
    A compactification of the Chern-Weil theory for bundle maps developed by Harvey and Lawson is described. For each smooth section $\nu$ of the compactification $\IP(\underline{\doubc}\oplus F)\to X$ of a rank n complex vector bundle $F\to X$ with connection, and for each Ad-invariant polynomial $\phi$ on ${\bf gl}\sb{n},$ there are associated current formulae generalizing those of Harvey and Lawson. These are of the form $$\eqalign{\phi(\Omega\sb{\it F}) &+\rm \nu\sp*(Res\sb\infty(\phi))\ Div\sb\infty(\nu) - \phi(\Omega\sb0)\ -\cr&\qquad\rm Res\sb0(\phi)\ Div\sb0(\nu) = {\it dT}\quad on\ {\it X},\cr}$$where Div$\sb0(\nu)$ and Div$\sb\infty(\nu)$ are integrally flat currents supported on the zero and pole sets of $\nu,$ where Res$\sb0(\phi)$ and Res$\sb\infty(\phi)$ are smooth residue forms which can be calculated in terms of the curvature $\Omega\sb{F}$ of F, where T is a canonical transgression form with coefficients in $L\sbsp{\rm loc}{1},$ and where $\phi(\Omega\sb0)$ is an $L\sbsp{\rm loc}{1}$ form canonically defined in terms of a singular connection naturally associated to $\nu.$ These results hold for $C\sp\infty$-meromorphic sections $\nu$ which are atomic. The notion of an atomic section of a vector bundle was first introduced and studied by Harvey and Semmes. The formulae obtained include a generalization of the Poincare-Lelong formula to $C\sp\infty$-meromorphic sections of a bundle of arbitrary rank. Analogous results hold for real vector bundles and for quaternionic line bundles.
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    Integral Formulae Connected by Dolbeault's Isomorphism
    (Rice University, 1970-04) Harvey, F. Reese; Electronic version made possible with funding from the Rice Historical Society and Thomas R. Williams, Ph.D., class of 2000.
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    Kernels for d and (dee-bar)
    (1976) Graham, Charles Robin; Harvey, F. Reese
    This thesis studies kernels related to the d and a equations in Euclidean space. A formalism is developed for representing kernels on differential forms in an open set as currents on the product x n and the use of this formalism in studying d and a is investigated. Specific examples of kernels are analyzed in detail and their advantages and disadvantages with regard to possible applications are discussed.
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    On the boundaries of special Lagrangian submanifolds
    (1995) Fu, Lei; Harvey, F. Reese
    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.
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    Partial umbilics of hypersurfaces and repeated eigenvalue currents
    (2003) Earles, Christopher Michael; Harvey, F. Reese
    We present the theory of atomic sections developed by Harvey and Lawson, and we use it to study the repeated eigenvalue currents of symmetric bilinear forms. The main example is the classical theorem of surface theory which equates the total index of the umbilic points to the Euler characteristic of a compact surface in R3 . We derive this from the Harvey-Lawson viewpoint and extend it to surfaces with boundary. To develop analogous results for hypersurfaces in R2n+1 , we first prove a Splitting Principle for the differential characters of an oriented, even rank, real vector bundle and use it to compute the Euler character of the bundle of traceless symmetric bilinear forms. Finally, we show that partial umbilics of even type are boundaries with the single exception of the partial umbilics of type (2,...,2) (with n twos), which represent a multiple of the Euler class.