Scalar curvature rigidity for products of convex hypersurfaces

Abstract

Let $N=N_1\times \cdots \times N_k$, where each $N_i\subset {\mathbb{R}}^{n_i+1}$ is a closed strictly convex hypersurface. Let $M$ be a Riemannian spin manifold of dimension $n=\dim (N)$, and let $f:M\to N$ be an area non-increasing smooth map of non-zero degree. We show that ${\mathrm{scal}}_M\ge {\mathrm{scal}}_N\circ f$ implies ${\mathrm{scal}}_M={\mathrm{scal}}_N\circ f$. Moreover, if $n\ge 3$ and $N$ has no circle factors, then every such map is a Riemannian isometry. In the presence of circle factors, we obtain the corresponding optimal splitting theorem for $M$ and $f$. Our results are based on an approach to the index-theoretic part of Llarull's scalar curvature rigidity theorem via Clifford-linear family index theory, which works independently of the parity of the dimension and extends naturally to products. This includes a proof of the Geroch conjecture for spin manifolds as the edge case with only circle factors.

15 pages