Diffeomorphism Stability and Codimension Three

2016/06/06 by Pro, Curtis, Wilhelm, Frederick
#53C20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1606.01828

Abstract

Given k∈ ℝ, v, D>0, and n∈ ℕ, let \ M_α\ _α=1^∞ be a Gromov-Hausdorff convergent sequence of Riemannian n--manifolds with sectional curvature ≥ k, volume >v, and diameter ≤ D. Perelman's Stability Theorem implies that all but finitely many of the M_αs are homeomorphic. The Diffeomorphism Stability Question asks whether all but finitely many of the M_αs are diffeomorphic. We answer this question affirmatively in the special case when all of the singularities of the limit space occur along smoothly and isometrically embedded Riemannian manifolds of codimension ≤ 3. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension at ≤ 3, then all but finitely many of the M_αs are diffeomorphic.

Related

• Perelman's Stability Theorem
• Limits of manifolds with boundary II
• Proving homological stability for homeomorphisms of manifolds
• A note on the topological stability theorem from RCD spaces to Riemannian manifolds
• Gromov-Hausdorff Limits of Aspherical Manifolds