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Zhenhua Liu: Lectures

May 17, 2024
TITLE: General behavior of area-minimizing submanifolds

ABSTRACT: We will review some recent progress on the general geometric behavior of homologically area-minimizing currents, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as special Lagrangians on a Calabi-Yau, etc. A fine understanding of the geometric structure of homological area-minimizers can give far-reaching consequences for related problems. Camillo De Lellis and his collaborators have proven that singular sets of area-minimizing integral currents are codimension-two rectifiable. A pressing next question is what one can say about the geometric behavior of area-minimizing currents beyond this. Almost all known examples and results point towards that area-minimizing subvarieties are subanalytic, generically smooth, calibrated, and have a priori curvature bounds. It is natural to ask if these hold in general. In this direction, we prove that all of these properties thought to be true generally and proven to be true in special cases are totally false in general. We prove that smoothly calibrated area-minimizing subvarieties can have fractal singular sets. Smoothable singularities are non-generic. Calibrated area minimizers are non-generic. A priori L^2 curvature bounds even fail for holomorphic subvarieties. Consequently, we answer several conjectures of Frederick J. Almgren Jr., Frank Morgan, and Brian White from the 1980s.

September 14, 2021
TITLE: Every finite graph arises as the singular set of a compact 3-d calibrated area minimizing submanifold

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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