Fourth order curvature flows and geometric applications

TL;DR

This paper investigates fourth order curvature flows on compact manifolds, showing under certain conditions they avoid collapsing singularities and converge to space forms, providing new insights into geometric analysis and stability of these flows.

Contribution

It demonstrates that with a positive Yamabe invariant lower bound, certain fourth order curvature flows cannot collapse and must converge to standard space forms, extending previous results.

Findings

Flows only develop curvature blow-up singularities under certain conditions.

Sequences of blow-up limits are Bach-flat, scalar-flat manifolds.

Flow convergence to sphere or real projective space under energy bounds.

Abstract

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.