Time：2019-02-18 14:30-15:30
2019-02-18 16:00-17:00
2019-02-19 11:00-12:00
Location：Conference Room 108 at Experiment Building at Haiyun Campus
Abstract: Minimal surfaces become stationary solutions of the mean curvature flow mathematical models for Plateau’s soap film experiments, as natural higher dimensional generalization of geodesics. The modern theory of minimal surfaces offers spectacular applications to, for instance, the three dimensional topology geometry (by Meeks, Scheon, Simon, Yau), positive mass conjecture in mathematical relativity (by Scheon, Yau), the Ricci flow proof of Poincare’s conjecture (by Perelman). In this mini-course, we present a comprehensive introduction to some of important global results interesting examples in the minimal surface theory. We sketch several proofs of Bernstein's beautiful theorem that the only entire minimal graphs in Euclidean three-space are flat planes, present various Bernstein type results in Euclidean four-space, which highlight the role of complex analysis in the modern theory of minimal surfaces. Inspired by the Finn-Osserman (1964), Chern (1969), do Carmo-Peng (1979) proofs of the Bernstein theorem, we prove a new rigidity theorem for associate families connecting the doubly periodic Scherk graphs the singly periodic Scherk towers. Our characterization of Scherk's surfaces discovers a new idea from the original Finn-Osserman curvature estimate. Combining two generically independent flat structures introduced by Chern Ricci, we shall construct geometric harmonic functions on minimal surfaces, establish that periodic minimal surfaces admit fresh uniqueness results.
Recommended Reading List:
[1] Joaquin Perez, A new golden age of minimal surfaces, Notices Amer. Math.Soc. 64 (2017), no. 4, 347-358.
[2] Jeremy Gray, Mario Micallef, About the cover: the work of Jesse Douglas on minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 293-302.
