Integrable Geometry: Curves, Surfaces and Flows | |
Student No.： | 20 |
Time： | Wed & Fri 15:20-16:55 |
Instructor： | Franz Pedit [University of Tuebingen, Germany; University of Massachusetts, Amherst, USA ] |
Place： | Conference Room 2, Jin Chun Yuan West Bldg. |
Starting Date： | 2018-4-11 |
Ending Date： | 2018-6-20 |
Description：
This course will give an introduction to the structure of the space of all closed curves and surfaces. These spaces carry a natural symplectic structure and Hamiltonian functions. For curves one has length, enclosed area, total torsion, the bending energy etc. For surfaces one has conformal type, Dirichlet energy, enclosed volume, the Willmore energy and so on. The stationary points of those functionals give important natural classes of curves and surfaces: elastic curves, constant mean curvature surfaces, Willmore surfaces, harmonic maps etc. The gradient and symplectic gradient flows of those functionals provide insight into the structure of the space of closed curves and compact surfaces in R^n. For example, the space of closed curves in R^3 is the phase space of the infinite dimensional integrable hierarchy related to the non-linear Schroedinger equation; constant mean curvature tori provide the phase space for the sinh-Gordon hierarchy; the Willmore gradient flow moves closed curves to closed elastica and immersed spheres of small enough Willmore energy to the round sphere. This course attempts to put these various phenomena into a coherent and somewhat unifying geometric setting.
Prerequisite： Basic knowledge of Differential Geometry and Riemann surfaces.
Reference：
G. Dziuk, E. Kuwert, R. Schatzle. Evolution of elastic curves in R^n: Existence and Computation, SIAM J. Math. Anal. 33 (2002), no. 5, 1228–1245.
D. Ferus; K. Leschke, F. Pedit, U. Pinkall. Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori, Invent. Math.146 (2001), no. 3, 507–593.
N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Diff. Geom. 31 (1990), 627–710.
E. Kuwert, R. Schätzle. Gradient flow for the Willmore functional, Comm. Anal. Geom. 10 (2002), no. 2, 307–339.
F. Pedit and U. Pinkall. Quaternionic analysis on Riemann surfaces and differential geometry, Doc. Math. J. DMV, Extra Volume ICM 1998, Vol. II, 389-400.