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Title:
Every finite graph arises as the singular set of a compact 3-d calibrated area minimizing submanifold
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Abstract:
Calibrated submanifolds play an important role in special holonomic geometries. Prime examples include associative and coassociative submanifolds, special Lagrangian submanifolds, and Cayley subamnifolds. However, currently, we do not know much about the moduli space of such submanifolds. Just recall that a nice sequence of smooth algebraic curves can converge to a curve with branch points. Thus, it’s both inevitable and important to understand the singularities of calibrated submanifolds and how the singularities are formed. Now, Almgren’s Big Regularity Theorem and De Lellis-Spadaro’s new proof show that n-dimensional area minimizing integral currents are smooth manifolds outside of a singular set of dimensions at most n-2. Since calibrated submanifolds are area minimizing, we have this optimal bound on the dimension of their singular sets. A more geometrically applicable question is the fine structure of the singular set. The problem has been settled in dimension 2 by Chang and De Lellis-Spadaro-Spolaor, and they prove that all 2-dimensional area minimizing currents are branched with minimal immersion in the interior. Starting from dimension 3, very little is known. In particular, it is not even known if a line segment can appear as the singular set of a 3-d calibrated submanifold. In this direction, we show that given any (not necessarily connected) finite graph in the combinatorial sense, we can construct a calibrated 3-dimensional calibrated submanifold on a 7-dimensional closed compact Riemannian manifolds, so that the singular set of the surfaces consist of precisely this finite graph. The calibration is a modification of the special Lagrangian form. Both the metric and the calibration form are smooth. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.
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