Fourth-order geometric flows and integral pinching of the curvature
type: Dataset
description: We study fourth-order geometric flows on compact Riemannian manifolds, which naturally appear as gradient flows of quadratic curvature functionals. When the Yamabe constant remains bounded from below by a positive constant along the flow, we show that the manifold doesn't collapse, and that a sequence of dilated metrics near a singular time converges to a singularity model. In particular, in dimension four, this assumption is satisfied by a class of gradient flows, provided that the initial energy is less than an explicit constant. The singularities of these flows are then modeled by complete non-compact manifolds, which are Bach-flat and scalar-flat. By combining a Weitzenböck formula with the Sobolev inequality induced by the positivity of the Yamabe constant, we prove several rigidity results for metrics with integral pinched curvature. In particular, we prove a rigidity result for Bach-flat and scalar-flat manifolds in dimension four, which implies that the singularities of our gradient flows can only exist when the initial energy is bigger than a given constant. When this is not the case, these flows exist for all time, and converge to a metric with constant positive curvature. It provides a proof of a "sphere theorem" for closed four-dimensional manifolds with integral pinched curvature. Applying the same method to harmonic forms on an integral pinched manifold, we prove an integral version of the Bochner-Weitzenböck theorem. As a corollary, we obtain the vanishing of Betti numbers under various integral pinching conditions, and we characterize the equality cases.
landing page: https://theses.hal.science/tel-00771720
identifier: NNT: 2012GRENM040
keyword: bochner-technique
keyword: equations-dordre-quatre
keyword: flots-geometriques
keyword: fourth-order-equations
keyword: geometric-flows
keyword: geometrie-riemannienne
keyword: infoeu-reposemanticsdoctoralthesis
keyword: integral-pinching
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: pincement-integral
keyword: riemannian-geometry
keyword: rigidite
keyword: rigidity
keyword: technique-de-bochner
keyword: theses
issued: 2026-05-15T11:44:12.165002
modified: 2026-05-15T11:44:12.165007
language: fr
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Fourth-order geometric flows and integral pinching of the curvature
type: Distribution
format: HTML
access URL: https://theses.hal.science/tel-00771720
issued: 2026-05-15T11:44:12.175298
modified: 2026-05-15T11:44:12.122639