Related papers: Generalized common index jump theorem with applica…

In this paper, we prove that for every dynamically convex compact star-shaped hypersurface $\Sigma\subset\mathbb{R}^{2n}$, there exist at least $\lfloor\frac{n+1}{2}\rfloor$ geometrically distinct closed characteristics possessing…

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact…

There is a long standing conjecture that there are at least $n$ closed characteristics for any compact convex hypersurface $\Sigma$ in $\mathbb{R}^{2n}$, and the symmetric case, i.e. $\Sigma=-\Sigma$, has already been proved by C. Liu, Y.…

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface $\Sigma\subset{\bf R}^{2n}$, there exist at least $n$ non-hyperbolic closed characteristics with even Maslov-type indices on $\Sigma$ when…

Let $\Sigma\subset \R^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface and only has finitely geometrically distinct closed characteristics. Based on Y.Long and C.Zhu 's index jump methods \cite{LoZ1}, we prove that there are at…

In [LZ02] published in Annals of Mathematics, Long and Zhu established originally the common index jump theorem (CIJT) for symplectic paths in 2002, which has played an important role in later studies on periodic solution orbits for…

In this paper, we firstly generalize some theories developed by I. Ekeland and H. Hofer in [EkH] for closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$ to star-shaped hypersurfaces. As applications, we use…

In this paper, we prove that if all closed characteristics on a compact non-degenerate star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{2n}$ are elliptic, then either there exist exactly $n$ geometrically distinct closed characteristics,…

Resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding Hamiltonian systems. In this paper, we establish two new resonance identities…

In this paper, we prove that for every non-degenerate $C^3$ compact star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{6}$ which carries no prime closed characteristic of Maslov-type index $0$ or no prime closed characteristic of Maslov-type…

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in ${\bf R}^4$. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof.…

So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ are symmetric. In order to understand behaviors of such orbits, in this paper we establish first…

Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical…

In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing…

Let $\Sigma$ be a compact convex hypersurface in ${\bf R}^{2n}$ which is P-cyclic symmetric, i.e., $x\in \Sigma$ implies $Px\in\Sigma$ with P being a $2n\times2n$ symplectic orthogonal matrix and $P^k=I_{2n}$, where $n, k\geq2$,…

We establish multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles. The results hold under certain index requirements on the contact form…

In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean…

We prove that any $n$-dimensional closed mean convex $\lambda$-hypersurface is convex if $\lambda\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $\lambda$-hypersurfaces. As a corollary, we obtain a gap theorem…

Let $\Sigma\subset \mathbb{R}^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface. The stability of closed characteristics has attracted considerable attention in related research fields. A long-standing conjecture states that all…

In this paper, we proved that for every non-degenerate $C^3$ compact star-shaped hypersurface $\Sigma$ in $\mathbb{R}^{8}$ which carries no prime closed characteristic of Maslov-type index $-1$, there exist at least four prime closed…