Flows of geometric structures

Autor(es)

Fadel, Daniel ; Loubeau, Eric ; Moreno, Andres J. ; Sa Earp, Henrique N.

• Tipo de documento: Artigo Científico
• Fonte: JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK; v. N/A, p. 86-pg., 2024-09-03.

Resumo

We develop an abstract theory of flows of geometric H-structures, i.e., flows of tensor fields definingH-reductions of the frame bundle, for a closed and connected subgroup H SO.n/, on any connected and oriented n-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of H-structures, by way of the natural infinitesimal action of GL.n; R/ on tensors combined with various bundle decompositions induced byH-structures. We compute evolution equations for the intrinsic torsion under general flows of H-structures and, as applications, we obtain general Bianchi-type identities for H-structures, and, for closed manifolds, a general first variation formula for the L2-Dirichlet energy functional E on the space of H-structures. We then specialise the theory to the negative gradient flow of E over isometric H-structures, i.e., their harmonic flow. The core result is an almost-monotonocity formula along the flow for a scaleinvariant localised energy, similar to the classical formulas by Chen-Struwe [M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), no. 3, 485-502; Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), no. 1, 83-103] for the harmonic map heat flow. This yields an "-regularity theorem and an energy gap result for harmonic structures, as well as longtime existence for the flow under small initial energy, relative to the L1 -norm of initial torsion, in the spirit of Chen-Ding [Y. M. Chen and W. Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), no. 3, 567-578]. Moreover, below a certain energy level, the absence of a torsion-free isometric H-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat n-tori, so long as the set OESn; SO. n/=H _ of homotopy classes of maps Sn ! SO. n/=H contains more than one element and the universal cover of SO. n/=H is a sphere, e.g. when n D 7 and H D G2, or n D 8 and H D Spin.7/. (AU)

• Processo FAPESP: 21/04065-6 - BRIDGES: interações França-Brasil em Teoria de Calibres, estruturas extremais e estabilidade
• Beneficiário: Henrique Nogueira de Sá Earp
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 18/21391-1 - Teoria de calibre e geometria algébrica
• Beneficiário: Marcos Benevenuto Jardim
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 21/08026-5 - Geometrias especiais e subvariedades calibradas
• Beneficiário: Andrés Julián Moreno Ospina
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado