Scalar Curvature Question [?46]: But it is unclear if this remain true with "area" in place of "length"

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.

Let $(Y,g_0)$ be an oriented Riemannian $n$-manifold. In the degree comparison consider oriented Riemannian $n$-manifolds $(X,g)$ and proper maps $f:X\to Y$ of nonzero degree. Proper means that inverse images of compact sets are compact; degree is the integer multiplying the orientation fundamental class (the relative class when the map sends boundary to boundary). A map is area non-increasing when the area of each smooth parametrized surface is not increased by its composition with $f$, with multiplicity counted. For a Lipschitz map, area is computed from its almost-everywhere differential. Scalar curvature $\operatorname{Sc}_g$ is the trace of the Ricci curvature tensor. The smooth area-extremality assertion here excludes all $C^2$ maps in this degree class satisfying both area non-increase and $\operatorname{Sc}_g(x)>\operatorname{Sc}_{g_0}(f(x))$ for every $x\in X$.

Statement recorded in the supplied source.

If this smooth area-extremality assertion holds for $(Y,g_0)$, must it still hold when $C^2$ maps are replaced by Lipschitz maps in the same proper, nonzero-degree comparison class? [S2]

Short English statement

If no smooth map of nonzero degree can decrease areas while sending an everywhere higher-scalar-curvature manifold onto the target, is the same exclusion valid for Lipschitz maps?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6700043 (AMR-066-0043). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Misha Gromov, A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature (18 October 2017), Section 4, printed pp. 10--14; arXiv:1710.05946v1. https://arxiv.org/pdf/1710.05946v1.