Curve shortening flows and applications

• Grant number: 17/19725-6
• Support Opportunities: Regular Research Grants
• Duration: February 01, 2018 - July 31, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Joachim Weber
• Grantee: Joachim Weber
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

Consider a closed Riemann surface. Our ultimate goal is to construct Morse homology groups associated to the length functional L on the space C of unparametrized immersed loops. Namely, equivalence classes under reparametrization of parametrized immersed loops. The Morse chain groups are generated by the critical points of L - the periodic geodesics of the Riemann surface - and graded by the Morse index. The boundary operator counts flow lines of the curve shortening flow (CSF) between two periodic geodesics of index difference one. Such construction is motivated by - and eventually aims to refine - Angenent's work. The aim of the present research project is more modest: To avoid finite time singularities under CSF we restrict to the part E of C that consists of embedded unparametrized loops. We aim to construct Morse homology groups HM(CSF(E)) for the L2 gradient of the length functional L on the space E, but relative to the exit set of short loops. Adapting suitably recent Conley index methods to construct a Morse filtration we aim to establish a natural isomorphism to relative singular homology. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

FRAUENFELDER, URS; WEBER, JOA. The fine structure of Weber's hydrogen atom: Bohr-Sommerfeld approach. ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, v. 70, n. 4, AUG 2019. (17/19725-6)

FRAUENFELDER, URS; WEBER, JOA. The shift map on Floer trajectory spaces. Journal of Symplectic Geometry, v. 19, n. 2, p. 351-397, 2021. (17/19725-6)