RESEARCH
The multiplicity and stability conjectures about closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$

2020-11-19　10:00 — 11:30

There are two long standing conjectures on the multiplicity and stability of closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$ since the time of Liapunov in 1892:

(1) Every compact convex hypersurface $\Sigma$ in ${\bf R}^{2n}$ carries at least n closed characteristics.

(2) There always exists an elliptic closed characteristic on any compact convex hypersurface  $\Sigma$ in ${\bf R}^{2n}$.

Since the breakthroughs of Y. Long and C. Zhu in 2002, the multiplicity conjecture has been solved by W. Wang-X. Hu-Y. Long and W. Wang for the cases n=3 and n=4, respectively. For general dimension case,   G.Dell’Antonio-B. D’Onofrio-I. Ekeland and C. Liu-Y. Long-C. Zhu solved the stability and multiplicity conjectures, respectively if $\Sigma$ is centrally symmetric. In this talk, I will discuss our recent progresses on the two conjectures for compact convex hypersurfaces with more general symmetries.