Cohomogeneity one solitons for the isometric flow of -structures

Area of Science:

• Differential Geometry
• Geometric Analysis
• Topology

Background:

• Isometric flow is a fundamental concept in geometric analysis, used to study the evolution of Riemannian metrics.
• Cohomogeneity one manifolds are important examples in geometry, often featuring special holonomy.
• Solitons are special solutions to geometric flows that remain invariant under the flow, providing insights into singularity formation.

Purpose of the Study:

• To investigate the existence of cohomogeneity one solitons for the isometric flow of G2-structures.
• To analyze the asymptotic behavior of torsion for these solitons.
• To establish the existence of shrinking isometric solitons and their implications for singularity formation.

Main Methods:

• The study employs techniques from Riemannian geometry and the analysis of nonlinear ordinary differential equations (ODEs).
• Specific focus is given to ODEs with regular singular points.
• General Riemannian geometric formulas for cohomogeneity one metrics on vector bundles are derived and applied.

Main Results:

• Global solutions to the isometric soliton equations are established for Euclidean R7, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and Bryant-Salamon G2-manifolds.
• The asymptotic behavior of torsion is determined in all considered cases.
• The existence of shrinking isometric solitons on R7 is proven, suggesting the likely presence of type I singularities.

Conclusions:

• The existence of cohomogeneity one solitons is confirmed across diverse G2-manifold settings.
• The analysis of associated nonlinear ODEs provides a robust framework for understanding soliton behavior.
• These findings contribute to the understanding of geometric flows and singularity formation in Riemannian geometry.