2010/12/01 by Vincent Bour, Bour, Vincent · 5 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG
paper · pdf · doi:10.48550/arxiv.1012.0342
arxiv created 2010/12/01 · openalex publication_date 2010/12/01 · arxiv updated 2010/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a class of fourth order curvature flows on a compact Riemannian manifold, which includes the gradient flows of a number of quadratic geometric functionals, as for instance the L2 norm of the curvature. Such flows can develop a special kind of singularities, that could not appear in the Ricci flow, namely singularities where the manifold collapses with bounded curvature. We show that this phenomenon cannot occur if we assume a uniform positive lower bound on the Yamabe invariant. In particular, for a number of gradient flows in dimension four, such a lower bound exists if we assume a bound on the initial energy. This implies that these flows can only develop singularities where the curvature blows up, and that blowing-up sequences converge (up to a subsequence) to a "singularity model", namely a complete Bach-flat, scalar-flat manifold. We prove a rigidity result for those model manifolds and show that if the initial energy is smaller than an explicit bound, then no singularity can occur. Under those assumptions, the flow exists for all time, and converges up to a subsequence to the sphere or the real projective space. This gives an alternative proof, under a slightly stronger assumption, of a result from Chang, Gursky and Yang asserting that integral pinched 4-manifolds with positive Yamabe constant are space forms.