Mean curvature solitons in an extended Ricci flow background

• Grant number: 23/13921-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): April 01, 2024
• Effective date (End): March 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Luiz Roberto Hartmann Junior
• Grantee: Matheus Hudson Gama dos Santos
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 22/16097-2 - Modern methods in differential geometry and geometric analysis, AP.TEM

Abstract

In this research project, we will consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying mean curvature flow in a Ricci flow background. Mainly, the functional we will focus on is the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We will compute its variational properties to the analysis of its time-derivative under Perelman's modified extended Ricci flow. In this time-derivative formula, one expects an extension of Hamilton's differential Harnack expression on the boundary integrand appears. We also will work on the mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by an extended Ricci flow from the perspective introduced by Magni, Mantegazza and Tsatis in the study of mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow and we intend to establish Huisken's monotonicity-type formula in this setting. Finally, we will use methods of reducing EDPs to ODEs for constructing explicit examples for this theory.