Every finite graph arises as the singular set of a compact $3$-d calibrated area minimizing surface

Zhenhua Liu

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arXiv · 2106.03199

Every finite graph arises as the singular set of a compact $3$-d calibrated area minimizing surface

Zhenhua Liu

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Abstract

Given any (not necessarily connected) combinatorial finite graph and any compact smooth $6$-manifold $M^6$ with the third Betti number $b_3\not=0$, we construct a calibrated 3-dimensional homologically area minimizing surface on $M$ equipped in a smooth metric $g$, so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly $GL(6,\mathbb{R})$ twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.

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Zhenhua Liu. 2021-06-06. Every finite graph arises as the singular set of a compact $3$-d calibrated area minimizing surface. https://arxiv.org/abs/2106.03199

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