Lipschitz rigidity for scalar curvature

• DOI: 10.4171/jems/1514
• Publication Date: 2024-09-04T07:31:47Z
• Authors: Simone Cecchini; Bernhard Hanke; Thomas Schick
• Keywords: Mathematics - Differential Geometry; Mathematics - Metric Geometry; Differential Geometry (math.DG); FOS: Mathematics; Primary: 51F30, 53C23, 53C24, Secondary: 30C65, 53C27, 58J20; Metric Geometry (math.MG); 0101 mathematics; 01 natural sciences

Let

M

be a closed smooth connected spin manifold of even dimension

n

, let

g

be a Riemannian metric of regularity

W^{1,p}

,

p > n

, on

M

whose distributional scalar curvature in the sense of Lee–LeFloch is bounded below by

n(n-1)

, and let

f \colon (M,g) \to \mathbb{S}^n

be a

1

-Lipschitz continuous (not necessarily smooth) map of nonzero degree to the unit

n

-sphere. Then

f

is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his Four lectures . Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles. We argue that the existence of a nonzero harmonic spinor field forces

f

to be quasiregular in the sense of Reshetnyak, and in this way connect the powerful theory for quasiregular maps to the Atiyah–Singer index theorem.