Geometric Analysis and Topology Seminar

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Calibrated area-minimizing surfaces with fractal singular sets

Speaker: Zhenhua Liu, Princeton University

Location: Online

Videoconference link: https://nyu.zoom.us/j/94747124588

Date: Wednesday, January 26, 2022, 11 a.m.

Synopsis:

Calibrated surfaces appear naturally in many geometrical contexts, like holomorphic varieties or special Lagrangians. They are also area-minimizing, thus serving as the prime examples in geometric measure theory. However, just recall that a sequence of smooth holomorphic curves can converge to one with self-intersections and branch points. Singularities appear naturally when one considers moduli spaces of calibrated surfaces. Except for the 2-d case, where all area-minimizing surfaces are branched minimal immersions, we know very little beyond the codimension two Hausdorff dimension bound. For example, can the singular set of a 3-dimensional calibrated surface be the Cantor set? In this talk, we will discuss some recent constructions, and answer this question affirmatively. In fact, we can construct 3-d surfaces calibrated by smooth 3-forms on smooth compact Riemannian manifolds, so that the singular sets can be prescribed to be any closed subset of a closed interval, or even any closed subset of any finite combinatorial graph that contains at least some neighborhood of each vertex. Thus, any real number between 0 and 1 can be realized as the dimension of the singular set of a 3-d calibrated surface, and we can have a Cantor set worth of singularity. In general dimensions, we provide a sharp answer to a conjecture by Almgren.