Compact Star-shaped Hypersurfaces Research Articles

A dichotomy result for closed characteristics on compact star-shaped hypersurfaces in $$\mathbf{R}^{2n}$$

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, or there exist infinitely many geometrically distinct closed characteristics.

Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in $$\mathbf{R}^{2n}$$ R 2 n

In this paper, we firstly generalize some theories developed by Ekeland and Hofer (Commun Math Phys 113:419–467, 1987) for closed characteristics on compact convex hypersurfaces in $$\mathbf{R}^{2n}$$ to star-shaped hypersurfaces. As applications we use Ekeland–Hofer theory and index iteration theory to prove that if a compact star-shaped hypersuface in $$\mathbf{R}^4$$ satisfying some suitable pinching condition carries exactly two geometrically distinct closed characteristics, then both of them must be elliptic. We also conclude that the theory developed by Long and Zhu (Ann Math 155:317–368, 2002) still holds for dynamically convex star-shaped hypersurfaces, and combining it with the results in Wang et al. (Duke Math J 139:411–462, 2007), Liu et al. (J Funct Anal 266:5598–5638, 2014) and Wang (Adv Math 297:93–148, 2016), we obtain that there exist at least n closed characteristics on every dynamically convex star-shaped hypersurface in $$\mathbf{R}^{2n}$$ for $$n=3, 4$$ .

Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ℝ2n

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface Σ ⊂ R2n, there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on Σ when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on Σ and at least (n−1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface Σ ⊂ R2n index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (τ, y) on Σ possesses positive mean index and whose Maslov-type index i(y,m) of its m-th iterate satisfies i(y,m) ≠ −1 when n is even, and i(y,m) ∉ {−2,−1, 0} when n is odd for all m ∈ N.

Irrationally elliptic closed characteristics on symmetric compact star-shaped hypersurfaces in $$\mathbf{R}^\mathbf{4}$$ R 4

Irrationally elliptic closed characteristics on symmetric compact star-shaped hypersurfaces in $$\mathbf{R}^\mathbf{4}$$ R 4

Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

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 two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ when there exist only finitely many geometrically distinct symmetric closed characteristics on $\Sigma$, which extend the identity established by Liu and Long in \cite{LLo1} of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu, Long and Wang recently in \cite{LLW1} for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in ${\bf R}^4$, then both of them must be elliptic.

The existence of two closed characteristics on every compact star-shaped hypersurface in ℝ4

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in ℝ4. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.

Resonance identities for closed characteristics on compact star-shaped hypersurfaces in [formula omitted

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 for closed characteristics on every compact star-shaped hypersurface Σ in R2n when the number of geometrically distinct closed characteristics on Σ is finite, which extend those identities established by C. Viterbo in 1989 for star-shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non-degenerate, and that by W. Wang, X. Hu and Y. Long in 2007 for strictly convex hypersurfaces in R2n.

Uniqueness of starshaped compact hypersurfaces with prescribed $m$-th mean curvature in hyperbolic space

Let $\psi$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $\psi$, when evaluated on $M$, coincides with the $m$-th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980s. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigate the problem of uniqueness in hyperbolic space and show that uniqueness (up to a geometrically trivial transformation) holds under the same conditions under which existence was established.