Scalar Curvature Question [?7]: But deeper structures (if they exist at all) that lie at the roots of Dirac operators and of minimal hypersurf

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Dirac operator is a first-order elliptic operator on spinor fields, or on spinors twisted by an auxiliary bundle, used to extract scalar-curvature constraints through its index and square. A minimal hypersurface is a submanifold of codimension one with vanishing mean curvature, equivalently stationary area under local variations. Minimal varieties in the source allow the corresponding higher-dimensional and singular geometric-measure-theory setting. Both techniques supply restrictions on scalar curvature, but the question does not prescribe a particular common structure or an equivalence theorem.

Statement recorded in the supplied source.

Identify deeper structures, if any, underlying both Dirac-operator methods and minimal-hypersurface methods in scalar curvature. Is there a direct link between Dirac operators and minimal varieties, or is their joint appearance in this setting accidental? [S2]

Short English statement

Seek a structural explanation, if one exists, for why Dirac operators and minimal hypersurfaces both govern scalar-curvature phenomena.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6700007 (AMR-066-0007). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Misha Gromov, Four Lectures on Scalar Curvature (2021 version). Section 3.20, Question 2 and its surrounding discussion, p. 204. https://arxiv.org/pdf/1908.10612v6.