Scalar-mean rigidity theorem and Llarull's theorem for nonspin manifolds

We prove Llarull’s scalar curvature rigidity theorem for spheres and scalar-mean rigidity for strictly convex Euclidean domains in all dimensions without the spin assumption.

Main theorem

Spherical rigidity without spin

Llarull's theorem in all dimensions

Let $(M^n,g)$ be a smooth, connected, closed Riemannian manifold with $n\geq2$, and let $F:M\to\mathbb S^n$ be a smooth map of nonzero degree to the unit round sphere. If $F$ is $1$-Lipschitz and $\operatorname{Sc}_g\geq n(n-1)$, then $F$ is an isometry. No spin assumption is required.

Proof architecture

The proof reduces spherical rigidity to a weighted scalar-mean comparison problem on complete manifolds, then descends in dimension while controlling the singularities of the capillary hypersurfaces.

Step 1 · Reduction

From the sphere to weighted scalar-mean comparison

Assuming rigidity fails, conformal deformations on a cylinder over $M$ produce a complete manifold with uniformly positive scalar curvature and a strict boundary mean curvature gap above the trace norm of the comparison map. The map retains nonzero degree. A weighted formulation of this comparison provides the curvature conditions needed for dimension descent.

Step 2 · Capillary bubbles

Pass to a hypersurface and obtain a spectral inequality

A barrier potential controls the noncompact end and produces a compact minimizing weighted capillary $\mu$-bubble. Its stability inequality yields a spectral weighted scalar-mean comparison in one lower dimension. The remaining obstruction to iteration is that the bubble may have interior or boundary singularities.

Step 3 · Singularities and completeness

Blow up the singular set while retaining the degree

Assouad dimension estimates give quantitative control of the singular set across scales. They control a Green-function blow-up that sends the singularities to infinity and equips the regular part with a complete smooth conformal metric. The boundary comparison map is modified near the new ends while preserving the nonzero degree and the spectral comparison needed for induction.

Step 4 · Close the dimension descent

Recover pointwise inequalities and reach dimension two

A positive function obtained from the spectral inequality adjusts the weight and restores pointwise weighted scalar and boundary mean curvature inequalities. This reproduces the complete comparison problem in one lower dimension. Iteration reaches a two-dimensional bubble, where Gauss–Bonnet contradicts the retained nonzero degree, completing the comparison theorem and hence Llarull rigidity.