Fonctionne avec Indico
v3.2.9
In this talk, I will report on recent results of an ongoing collaboration with Éric Loubeau, Andrés Moreno and Henrique Sá Earp on the study of the harmonic flow of H-structures. This is the negative gradient flow of a natural Dirichlet-type energy functional on a given isometric class of H-structures on a closed Riemannian n-manifold, where H is any normal reductive Lie subgroup of SO(n). Using general Bianchi-type identities of H-structures, we are able to prove monotonicity formulas for scale invariant local versions of the energy, similar to the classic formulas proved by Struwe and Chen (1988-89) in the theory of harmonic map heat flow. In particular, we deduce a general epsilon-regularity result along the harmonic flow and, more importantly, we get long time existence and finite time singularity results in parallel to the classical results proved by Chen-Ding (1990) in harmonic map theory. Namely, we show that if the energy of the initial H-structure is small enough, depending on the C^0-norm of its torsion, then the harmonic flow exists for all time and converges to a torsion-free H-structure. Moreover, we prove that the harmonic flow of H-structures develops a finite time singularity if the initial energy is sufficiently small but there is no torsion-free H-structure in the homotopy class of the initial H-structure. Finally, based on the analogous work of He-Li (2021) for almost complex structures, we give a general construction of examples where the later finite-time singularity result applies on the flat n-torus for any H such that the n-th homotopy group of the quotient SO(n)/H is non-trivial; e.g. when n=7 and H=G2, or when n=8 and H=Spin(7).