On the harmonic flow of geometric structures

Início: 28/04/2023 17:00

Término: 28/04/2023 18:00

Palestrante: Daniel Fadel (IMPA)

E-mail do Palestrante: fadel.daniel@gmail.com

Resumo: 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 an isometric class of $H$-structures on a closed Riemannian $n$-manifold, where $H$ is the stabilizer in $mathrm{SO}(n)$ of a finite collection of tensors in $mathbb{R}^n$. 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. We then 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. In particular, 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, provided the $n$-th homotopy group of the quotient $mathrm{SO}(n)/H$ is non-trivial; e.g. when $n=7$ and $H=mathrm{G}_2$, or when $n=8$ and $H=mathrm{Spin}(7)$.