Llarull's Rigidity Theorem

• Llarull's Rigidity Theorem is a landmark result in Riemannian geometry establishing that a 1-Lipschitz map with nonzero degree forces a manifold with scalar curvature at least n(n−1) to be isometric to the round sphere.
• The theorem employs Dirac operator methods, the Lichnerowicz–Weitzenböck formula, and index theory to rigorously connect curvature lower bounds with geometric rigidity.
• Extensions to low regularity, weighted settings, and stability analyses reveal how slight perturbations in curvature affect rigidity, while open questions remain on relaxing the spin condition.

Llarull's Rigidity Theorem for Scalar Curvature refers to a striking set of results in Riemannian geometry connecting lower bounds on scalar curvature, 1-Lipschitz (non-expanding) maps between manifolds, and topological degree. These theorems establish the geometric rigidity of the round sphere as an extremal object under scalar curvature and map contraction constraints, and quantify the precise circumstances under which such rigidity is unique or fails.

1. Formulation of Llarull's Scalar Curvature Rigidity Theorem

Let $(M^n, g)$ be a closed (compact, connected, without boundary) smooth Riemannian spin manifold, and $(S^n, g_{\mathrm{std}})$ the standard unit $n$-sphere with constant sectional curvature $1$ (hence scalar curvature $n(n-1)$). Suppose

• $\operatorname{scal}_g(x) \geq n(n-1)$ for all $x \in M$,
• $f: (M, g) \to (S^n, g_{\mathrm{std}})$ is a smooth $1$-Lipschitz (i.e., non-expanding) map,
• $\deg(f) \ne 0$ (topological degree nonzero).

Conclusion: $f$ is a Riemannian isometry onto the round sphere, i.e., $g = g_{\mathrm{std}}$ via $f$ and $df$ is an isometry at each point (Baer et al., 7 Jul 2025).

This theorem was first established by Llarull (1998) via Dirac operator and index theory techniques. The result quantitatively realizes the intuition that round spheres are extremizers for the scalar curvature among all (spin) metrics that can be mapped 1-Lipschitz onto the round sphere by a map of nonzero degree.

2. Mathematical Framework and Proof Outline

2.1. 1-Lipschitz Maps and Jacobian Bounds

A map $f:(M,g)\to(N,h)$ is $L$-Lipschitz if $d_N(f(x), f(y)) \leq L\, d_M(x,y)$ for all $x, y \in M$. For $L=1$, this means $f$ is non-expanding; in particular, such maps strictly decrease or preserve lengths, areas, and volumes. For a.e.\ $x$, $|\det d_xf|\le 1$. This $1$-Lipschitz property is crucial for volume and scalar curvature comparison arguments.

2.2. Spin Structures and the Dirac Operator

A manifold $M^n$ is spin if its second Stiefel–Whitney class vanishes, $w_2(M) = 0$. This permits the construction of the spinor bundle $\Sigma M$ and the associated Dirac operator $D: C^\infty(\Sigma M) \to C^\infty(\Sigma M)$. The Lichnerowicz–Weitzenböck formula asserts $$D^2 = \nabla^*\nabla + \frac14\,\operatorname{scal}_g$$, providing a direct analytic link between scalar curvature and the analytical properties of spinors.

2.3. Degree of a Map

For oriented, closed, $n$-manifolds, a continuous map $f: M \to N$ induces $f_*: H_n(M; \mathbb Z) \to H_n(N;\mathbb Z)$; the degree, $\deg(f)$, is the integer characterizing this induced map, measuring how $M$ is wrapped onto $N$.

2.4. Proof Mechanism

The proof constructs the twisted Dirac operator associated to the bundle $\Sigma M \otimes f^*\Sigma S^n$. Under $1$-Lipschitz and scalar curvature lower bounds, the Weitzenböck formula and Bochner methods show $D^2$ is nonnegative, with kernel only if $df$ is everywhere an isometry. The Atiyah–Singer index theorem relates the index of $D$ to $\deg(f)$; thus, nonzero degree implies existence of a parallel spinor and forces rigidity: $f$ must be an isometry (Baer et al., 7 Jul 2025, Cecchini et al., 2022, Li et al., 2023).

3. Degree, Surjectivity, and Rigidity for Scalar and Ricci Curvature

3.1. Degree vs. Surjectivity

A central research question is whether the nonzero degree condition (global topological constraint) can be relaxed to mere surjectivity (weaker geometric constraint).

Results:

• For $n=2$ (Gauss curvature case), surjectivity suffices: if a $1$-Lipschitz, surjective map from a closed surface $(M, g)$ with $K_g \geq 1$ onto $(S^2, g_{\mathrm{std}})$ exists, then $f$ is an isometry.
• For $n\ge 3$, surjectivity is not sufficient. Counterexamples: for any $n\ge 3$ and any closed $M^n$ admitting a positive scalar curvature metric, one can construct a metric $g_M$ with $scal_{g_M} \ge S_0 >0$ and a smooth surjective $L$-Lipschitz map $f: (M, g_M) \to (S^n,g_{\mathrm{std}})$ of degree zero. The method involves attaching a thin, long cylindrical neck via surgery to produce surjectivity while maintaining scalar curvature bounds, but the map is not an isometry (Baer et al., 7 Jul 2025).

3.2. Ricci Curvature Analogue

Replacing scalar curvature lower bound $scal_g \ge n(n-1)$ with the Ricci curvature lower bound $\operatorname{Ric}_g \ge n-1$ unifies surjectivity and degree:

• For any dimension $n$, if $f: (M,g) \to S^n$ is surjective, $L$-Lipschitz, and $|\det d_xf|\le 1$ a.e., and if $\operatorname{Ric}_g\ge n-1$, then $(M, g)$ is isometric to the round sphere; if $f$ is $1$-Lipschitz, it is an isometry (Baer et al., 7 Jul 2025).

This demonstrates the qualitative difference in geometric control endowed by Ricci curvature versus scalar curvature.

4. Low Regularity, Generalizations, and Extensions

Llarull-type rigidity results have been extended to lower regularity settings and to various geometric generalizations:

4.1. Low Regularity Metrics and Maps

• For $W^{1,p}$ metrics ($p > n$), the theorem holds: if $g$ is an admissible $W^{1,p}$ metric whose distributional scalar curvature (Lee-LeFloch sense) satisfies $scal_g \ge n(n-1)$, and $f : (M, g) \to S^n$ is a $1$-Lipschitz, nonzero degree map, then $f$ is an isometry (Cecchini et al., 2022).
• Llarull’s rigidity persists for $L^\infty$-metrics on $S^n$ with finitely many point singularities and for Lipschitz maps of nonzero degree, provided scalar curvature and metric domination hypotheses are satisfied off the singular set (Chu et al., 30 May 2024).

4.2. Four-dimensional and Higher-dimensional Extensions

• In dimension $4$, the spin assumption can be dropped: if $(M^4, g)$ is closed, oriented, and $scal_g \ge 12$, and if $f: M^4 \to S^4$ is smooth, distance-nonincreasing, and of nonzero degree, then $f$ is an isometry (Cecchini et al., 20 Feb 2024).
• The proof uses variational techniques (μ-bubbles), three-sphere rigidity with coefficients, and coupled Ricci flow/harmonic map heat flow, obviating the need for a global spin structure in dimension 4.

4.3. Warped Products and Capillary μ-bubbles

Variants of the rigidity theorem have been proved for domains in 3-dimensional warped product spaces with log-concave warping, using capillary μ-bubble variational arguments and appropriate boundary mean curvature conditions, subsuming the round sphere as a special case (Chai et al., 6 Mar 2025).

4.4. Weighted Manifolds

The Llarull–type rigidity theorem extends to weighted manifolds $(M^n, g, e^{-P}\mathrm{d}V_g)$, with the P-scalar curvature $R_P(g) = R(g) + 2\Delta P - |\nabla P|^2$, and area-nonincreasing maps of nonzero degree: under $R_P(g) \ge n(n-1)$, isometry is forced and the density $P$ is shown to be constant (Zhou et al., 16 Nov 2025).

5. Stability and Quantitative Rigidity

The theorem admits quantitative stability extensions: under perturbations of the scalar curvature lower bound, the manifold must be $C^0$–close to the standard sphere away from a small "bad set" (Hirsch et al., 2023). Explicit estimates on the "good" region and the exceptional set depend on the deviation from the extremal scalar curvature and the size of the mapping degree. These results fully resolve Gromov’s “spherical stability problem” in all dimensions.

Specifics:

• If $R_g \geq n(n-1) - \varepsilon$ and $f$ is 1-Lipschitz of degree $\ne 0$, $g$ is $C^0$–close to the standard round metric away from a subset of small volume, with deficit proportional to $\sqrt{\varepsilon}$ (Hirsch et al., 2023).

For dimension 3, stability holds in the Sormani–Wenger Intrinsic Flat topology under suitable isoperimetric and curvature control (Allen et al., 2023).

6. Analytical and Topological Techniques

The Dirac operator with twisting by pullbacks of the spinor bundle on the sphere is the central analytical tool, relating the Bochner–Lichnerowicz–Weitzenböck curvature term directly to the scalar curvature comparison. The index, via the Atiyah–Singer theorem, is governed topologically by the mapping degree. Alternative approaches employ μ-bubble minimal hypersurfaces (especially in dimension 4), Ricci/harmonic map flows, and in odd dimensions, spectral flow techniques for families of Dirac-type operators (Li et al., 2023).

Spectral analogues replace the pointwise scalar curvature lower bound with spectral lower bounds for Schrödinger operators involving the Laplacian and scalar curvature, and drive the same rigidity via analytic inequalities (Chai et al., 2023).

7. Broader Significance and Open Problems

Llarull’s rigidity theorem stands as the paradigmatic scalar curvature rigidity result, in opposition to the more powerful Ricci and sectional curvature rigidity theorems (such as Bonnet–Myers and Toponogov). The precise role of the mapping degree, as well as the emergence of counterexamples when the degree condition is weakened, sharply demarcate the boundaries of scalar curvature control. The theorem’s extensions to low regularity, weighted settings, stability under perturbations, and non-spin manifolds have driven a general program to understand scalar curvature under topological and analytic constraints. Key open problems include removing the spin assumption in higher dimensions generally, exploring other targets beyond spheres, and fully characterizing the stability landscape beyond the $C^0$ and SWIF topologies (Baer et al., 7 Jul 2025, Hirsch et al., 2023, Allen, 3 Apr 2024).