| Full text | |
| Author(s): |
Fowdar, Udhav
;
Earp, Henrique N. Sa
Total Authors: 2
|
| Document type: | Journal article |
| Source: | JOURNAL OF GEOMETRIC ANALYSIS; v. 34, n. 6, p. 48-pg., 2024-06-01. |
| Abstract | |
We formulate the gradient Dirichlet flow of Sp ( 2 ) Sp ( 1 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Sp}(2)\textrm{Sp}(1)$$\end{document} -structures on 8-manifolds, as the first systematic study of a geometric quaternion-Kahler (QK) flow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{G}_2$$\end{document} case. We also draw several comparisons with Spin ( 7 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(7)$$\end{document} -structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via epsilon \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} -regularity.We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU ( 3 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SU}(3)$$\end{document} . We also study compactness and the formation of singularities. (AU) | |
| FAPESP's process: | 21/07249-0 - Symmetries in exceptional holonomy problems |
| Grantee: | Udhav Fowdar |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 21/04065-6 - BRIDGES: Brazil-France interplays in Gauge Theory, extremal structures and stability |
| Grantee: | Henrique Nogueira de Sá Earp |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 18/21391-1 - Gauge theory and algebraic geometry |
| Grantee: | Marcos Benevenuto Jardim |
| Support Opportunities: | Research Projects - Thematic Grants |