Lekh Bhatia
Deformation of Curves on Surfaces: New Perspectives and Applications
This project will involve studying longtime behaviour of curve shortening flow in a variety of contexts. We begin by studying a proof of the established Gage-Hamilton-Grayson theorem using chord-arc compairson presented by Gerhard Huisken in 1998. We then extend these ideas in two main ways: the general case of mean curvature flow of hypersurfaces; the general case of curves with boundary conditions in convex domains rather than simply closed embedded curves. In doing so, we hope to identify a proof for the existence of at least two “Dirichlet-Neumann” geodesics from a point in a compact convex surface-with-boundary to the boundary. Herein lies the main research component of the project. We hope to identify a proof for this conjecture and possibly learn more about the longtime behaviour of curve shortening flow under boundary conditions.
Lekh Bhatia
Australian National University
Lekh Bhatia is an enthusiastic student from Sydney who recently completed his third year studying mathematics at the Australian National University. He has recently taken a keen interest in partial differential equations, particularly evolution equations in curvature flows. He has also held an interest in stochastic partial differential equations with applications to options pricing having worked in quantitative finance previously. He is preparing to pursue honours in geometric analysis next year. When Lekh is not studying mathematics, he enjoys rock climbing and listening to music.
