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

Abstract

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 $n=3, 4$.

Appendix

In the section, we briefly review the equivariant Morse theory and the resonance identities for closed characteristics on compact star-shaped hypersurfaces in $\mathbf{R}^{2n}$ developed in [22]. Now we fix a ${\Sigma }\in \mathcal{H}_{st}(2n)$ and assume the following condition:

(F) There exist only finitely many geometrically distinct prime closed characteristics $\quad \{(\tau _j, y_j)\}_{1\le j\le k}$ on $\Sigma$.

Let $\hat{\sigma }=\inf _{1\le j\le k}{\sigma _j}$ and T be a fixed positive constant. Then by Section 2 of [22], for any $a>\frac{\hat{\sigma }}{T}$, we can construct a function $\varphi _a\in C^{\infty }(\mathbf{R}, \mathbf{R}^+)$ which has 0 as its unique critical point in $[0, +\infty )$. Moreover, $\frac{\varphi ^{\prime }(t)}{t}$ is strictly decreasing for $t>0$ together with $\varphi (0)=0=\varphi ^{\prime }(0)$ and $\varphi ^{\prime \prime }(0)=1=\lim _{t\rightarrow 0^+}\frac{\varphi ^{\prime }(t)}{t}$. More precisely, we define $\varphi _a$ and the Hamiltonian function $\widetilde{H}_a(x)=a{\varphi }_a(j(x))$ via Lemma 2.2 and Lemma 2.4 in [22]. The precise dependence of $\varphi _a$ on a is explained in Remark 2.3 of [22].

For technical reasons we want to further modify the Hamiltonian, we define the new Hamiltonian function $H_a$ via Proposition 2.5 of [22] and consider the fixed period problem

$$\begin{aligned} \left\{ \begin{array}{lll} \dot{x}(t) &{}=&{} JH_a^\prime (x(t)), \\ x(0) &{}=&{} x(T). \\ \end{array}\right. \end{aligned}$$

(5.1)

Then $H_a\in C^{3}(\mathbf{R}^{2n} {\setminus }\{0\},\mathbf{R})\cap C^{1}(\mathbf{R}^{2n},\mathbf{R})$. Solutions of (5.1) are $x\equiv 0$ and $x=\rho z(\sigma t/T)$ with $\frac{{\varphi }_a^\prime (\rho )}{\rho }=\frac{\sigma }{aT}$, where $(\sigma , z)$ is a solution of (1.1). In particular, non-zero solutions of (5.1) are in one to one correspondence with solutions of (1.1) with period $\sigma <aT$.

For any $a>\frac{\hat{\sigma }}{T}$, we can choose some large constant $K=K(a)$ such that

$$\begin{aligned} H_{a,K}(x) = H_a(x)+\frac{1}{2}K|x|^2 \end{aligned}$$

(5.2)

is a strictly convex function, that is,

$$\begin{aligned} (\nabla H_{a, K}(x)-\nabla H_{a, K}(y), x-y) \ge \frac{{\epsilon }}{2}|x-y|^2, \end{aligned}$$

(5.3)

for all $x, y\in \mathbf{R}^{2n}$, and some positive ${\epsilon }$. Let $H_{a,K}^*$ be the Fenchel dual of $H_{a,K}$ defined by

$$\begin{aligned} H_{a,K}^*(y) = \sup \left\{ x\cdot y-H_{a,K}(x)\;|\; x\in \mathbf{R}^{2n}\right\} . \end{aligned}$$

The dual action functional on $X=W^{1, 2}(\mathbf{R}/{T \mathbf{Z}}, \mathbf{R}^{2n})$ is defined by

$$\begin{aligned} F_{a,K}(x) = \int _0^T{\left[ \frac{1}{2}(J\dot{x}-K x,x)+H_{a,K}^*(-J\dot{x}+K x)\right] dt}. \end{aligned}$$

(5.4)

Then $F_{a,K}\in C^{1,1}(X, \mathbf{R})$ and for $KT\not \in 2\pi \mathbf{Z}$, $F_{a,K}$ satisfies the Palais–Smale condition and x is a critical point of $F_{a, K}$ if and only if it is a solution of (5.1). Moreover, $F_{a, K}(x_a)<0$ and it is independent of K for every critical point $x_a\ne 0$ of $F_{a, K}$.

When $KT\notin 2\pi \mathbf{Z}$, the map $x\mapsto -J\dot{x}+Kx$ is a Hilbert space isomorphism between $X=W^{1, 2}(\mathbf{R}/{T \mathbf{Z}}; \mathbf{R}^{2n})$ and $E=L^{2}(\mathbf{R}/(T \mathbf{Z}),\mathbf{R}^{2n})$. We denote its inverse by $M_K$ and the functional

$$\begin{aligned} \Psi _{a,K}(u)=\int _0^T{\left[ -\frac{1}{2}(M_{K}u, u)+H_{a,K}^*(u)\right] dt}, \quad \forall \,u\in E. \end{aligned}$$

(5.5)

Then $x\in X$ is a critical point of $F_{a,K}$ if and only if $u=-J\dot{x}+Kx$ is a critical point of $\Psi _{a, K}$.

Suppose u is a nonzero critical point of $\Psi _{a, K}$. Then the formal Hessian of $\Psi _{a, K}$ at u is defined by

$$\begin{aligned} Q_{a,K}(v)=\int _0^T\left( -M_K v\cdot v+H_{a,K}^{*\prime \prime }(u)v\cdot v\right) dt, \end{aligned}$$

(5.6)

which defines an orthogonal splitting $E=E_-\oplus E_0\oplus E_+$ of E into negative, zero and positive subspaces. The index and nullity of u are defined by $i_K(u)=\dim E_-$ and $\nu _K(u)=\dim E_0$ respectively. Similarly, we define the index and nullity of $x=M_Ku$ for $F_{a, K}$, we denote them by $i_K(x)$ and $\nu _K(x)$. Then we have

$$\begin{aligned} i_K(u)=i_K(x),\quad \nu _K(u)=\nu _K(x), \end{aligned}$$

(5.7)

which follow from the definitions (5.4) and (5.5). The following important formula was proved in Lemma 6.4 of [29]:

$$\begin{aligned} i_K(x) = 2n([KT/{2\pi }]+1)+i^v(x) \equiv d(K)+i^v(x), \end{aligned}$$

(5.8)

where the index $i^v(x)$ does not depend on K, but only on $H_a$.

By the proof of Proposition 2 of [28], we have that $v\in E$ belongs to the null space of $Q_{a, K}$ if and only if $z=M_K v$ is a solution of the linearized system

$$\begin{aligned} \dot{z}(t) = JH_a''(x(t))z(t). \end{aligned}$$

(5.9)

Thus the nullity in (5.7) is independent of K, which we denote by $\nu ^v(x)\equiv \nu _K(u)= \nu _K(x)$.

By Proposition 2.11 of [22], the index $i^v(x)$ and nullity $\nu ^v(x)$ coincide with those defined for the Hamiltonian $H(x)=j(x)^\alpha$ for all $x\in \mathbf{R}^{2n}$ and some ${\alpha }\in (1,2)$. Especially $1\le \nu ^v(x)\le 2n-1$ always holds.

We have a natural $S^1$-action on X or E defined by

$$\begin{aligned} \theta \cdot u(t)=u(\theta +t),\quad \forall \, \theta \in S^1, \, t\in \mathbf{R}. \end{aligned}$$

(5.10)

Clearly both of $F_{a, K}$ and $\Psi _{a, K}$ are $S^1$-invariant. For any $\kappa \in \mathbf{R}$, we denote by

$$\begin{aligned} \Lambda _{a, K}^\kappa= & {} \left\{ u\in L^{2}\left( \mathbf{R}/{T \mathbf{Z}}; \mathbf{R}^{2n}\right) \;|\;\Psi _{a,K}(u)\le \kappa \right\} , \end{aligned}$$

(5.11)

$$\begin{aligned} X_{a, K}^\kappa= & {} \left\{ x\in W^{1, 2}\left( \mathbf{R}/(T \mathbf{Z}),\mathbf{R}^{2n}\right) \;|\;F_{a, K}(x)\le \kappa \right\} . \end{aligned}$$

(5.12)

For a critical point u of $\Psi _{a, K}$ and the corresponding $x=M_K u$ of $F_{a, K}$, let

$$\begin{aligned} {\Lambda }_{a,K}(u)= & {} {\Lambda }_{a,K}^{\Psi _{a, K}(u)} = \left\{ w\in L^{2}\left( \mathbf{R}/(T\mathbf{Z}), \mathbf{R}^{2n}\right) \;|\; \Psi _{a, K}(w)\le \Psi _{a,K}(u)\right\} , \end{aligned}$$

(5.13)

$$\begin{aligned} X_{a,K}(x)= & {} X_{a,K}^{F_{a,K}(x)} = \left\{ y\in W^{1, 2}\left( \mathbf{R}/(T\mathbf{Z}), \mathbf{R}^{2n}\right) \;|\; F_{a,K}(y)\le F_{a,K}(x)\right\} . \end{aligned}$$

(5.14)

Clearly, both sets are $S^1$-invariant. Denote by $\mathrm{crit}(\Psi _{a, K})$ the set of critical points of $\Psi _{a, K}$. Because $\Psi _{a,K}$ is $S^1$-invariant, $S^1\cdot u$ becomes a critical orbit if $u\in \mathrm{crit}(\Psi _{a, K})$. Note that by the condition (F), the number of critical orbits of $\Psi _{a, K}$ is finite. Hence as usual we can make the following definition.

Definition 5.1

Suppose u is a nonzero critical point of $\Psi _{a, K}$, and $\mathcal{N}$ is an $S^1$-invariant open neighborhood of $S^1\cdot u$ such that $\mathrm{crit}(\Psi _{a,K})\cap ({\Lambda }_{a,K}(u)\cap \mathcal{N}) = S^1\cdot u$. Then the $S^1$-critical modules of $S^1\cdot u$ are defined by

$$\begin{aligned} C_{S^1,\; q}\left( \Psi _{a, K}, \;S^1\cdot u\right) =H_{q}\left( \left( \Lambda _{a, K}(u)\cap \mathcal{N}\right) _{S^1},\; \left( \left( \Lambda _{a,K}(u){\setminus } S^1\cdot u\right) \cap \mathcal{N}\right) _{S^1}\right) . \end{aligned}$$

Similarly, we define the $S^1$-critical modules $C_{S^1,\; q}(F_{a, K}, \;S^1\cdot x)$ of $S^1\cdot x$ for $F_{a, K}$.

We fix a and let $u_K\ne 0$ be a critical point of $\Psi _{a, K}$ with multiplicity $\mathrm{mul}(u_K)=m$, that is, $u_K$ corresponds to a closed characteristic $(\tau , y)\subset \Sigma$ with $(\tau , y)$ being m-iteration of some prime closed characteristic. Precisely, we have $u_K=-J\dot{x}+Kx$ with x being a solution of (5.1) and $x=\rho y(\frac{\tau t}{T})$ with $\frac{{\varphi }_a^\prime (\rho )}{\rho }=\frac{\tau }{aT}$. Moreover, $(\tau , y)$ is a closed characteristic on $\Sigma$ with minimal period $\frac{\tau }{m}$. By Lemma 2.10 of [22], we construct a finite dimensional $S^1$-invariant subspace G of $L^{2}(\mathbf{R}/{T \mathbf{Z}}; \mathbf{R}^{2n})$ and a functional $\psi _{a,K}$ on G. For any $p\in \mathbf{N}$ satisfying $p\tau <aT$, we choose K such that $pK\notin \frac{2\pi }{T}\mathbf{Z}$, then the pth iteration $u_{pK}^p$ of $u_K$ is given by $-J\dot{x}^p+pKx^p$, where $x^p$ is the unique solution of (5.1) corresponding to $(p\tau , y)$ and is a critical point of $F_{a, pK}$, that is, $u_{pK}^p$ is the critical point of $\Psi _{a, pK}$ corresponding to $x^p$. Denote by $g_{pK}^p$ the critical point of $\psi _{a,pK}$ corresponding to $u_{pK}^p$ and let $\widetilde{\Lambda }_{a,K}(g_K)=\{g\in G \;|\; \psi _{a, K}(g)\le \psi _{a, K}(g_K)\}$.

Now we use the theory of Gromoll and Meyer, denote by $W(g_{pK}^p)$ the local characteristic manifold of $g_{pK}^p$. Then we have

Proposition 5.2

(cf. Proposition 4.2 of [22]) For any $p\in \mathbf{N}$, we choose K such that $pK\notin \frac{2\pi }{T}\mathbf{Z}$. Let $u_K\ne 0$ be a critical point of $\Psi _{a, K}$ with $\mathrm{mul}(u_K)=1$, $u_K=-J\dot{x}+Kx$ with x being a critical point of $F_{a, K}$. Then for all $q\in \mathbf{Z}$, we have

$$\begin{aligned}&C_{S^1,\; q}(\Psi _{a,pK},\;S^1\cdot u_{pK}^p) \nonumber \\&\quad \cong \left( \frac{}{}H_{q-i_{pK}(u_{pK}^p)}(W(g_{pK}^p)\cap \widetilde{\Lambda }_{a,pK}(g_{pK}^p),(W(g_{pK}^p) {\setminus }\{g_{pK}^p\})\cap \widetilde{\Lambda }_{a,pK}(g_{pK}^p))\right) ^{\beta (x^p)\mathbf{Z}_p}, \nonumber \\ \end{aligned}$$

(5.15)

where $\beta (x^p)=(-1)^{i_{pK}(u_{pK}^p)-i_K(u_K)}=(-1)^{i^v(x^p)-i^v(x)}$. Thus

$$\begin{aligned} C_{S^1,q}\left( \Psi _{a,pK},\;S^1\cdot u_{pK}^p\right) =0 \quad if q<i_{pK}\left( u_{pK}^p\right) ~ or ~q>i_{pK}\left( u_{pK}^p\right) +\nu _{pK}\left( u_{pK}^p\right) -1.\nonumber \\ \end{aligned}$$

(5.16)

In particular, if $u_{pK}^p$ is non-degenerate, i.e., $\nu _{pK}(u_{pK}^p)=1$, then

$$\begin{aligned} C_{S^1,\; q}(\Psi _{a,pK},\;S^1\cdot u_{pK}^p) = \left\{ \begin{array}{ll} \mathbf{Q}, &{} \quad {\mathrm{if}}\;q=i_{pK}\left( u_{pK}^p\right) \;{\mathrm{and}}\;\beta (x^p)=1, \\ 0, &{}\quad {\mathrm{otherwise}}. \\ \end{array}\right. \end{aligned}$$

(5.17)

We make the following definition:

Definition 5.3

For any $p\in \mathbf{N}$, we choose K such that $pK\notin \frac{2\pi }{T}\mathbf{Z}$. Let $u_K\ne 0$ be a critical point of $\Psi _{a,K}$ with $\mathrm{mul}(u_K)=1$, $u_K=-J\dot{x}+Kx$ with x being a critical point of $F_{a, K}$. Then for all $l\in \mathbf{Z}$, let

Here $k_l(u_{pK}^p)$’s are called critical type numbers of $u_{pK}^p$.

By Theorem 3.3 of [22], we obtain that $k_l(u_{pK}^p)$ is independent of the choice of K and denote it by $k_l(x^p)$, here $k_l(x^p)$’s are called critical type numbers of $x^p$.

We have the following properties for critical type numbers:

Proposition 5.4

(cf. Proposition 4.6 of [22]) Let $x\ne 0$ be a critical point of $F_{a,K}$ with $\mathrm{mul}(x)=1$ corresponding to a critical point $u_K$ of $\Psi _{a, K}$. Then there exists a minimal $K(x)\in \mathbf{N}$ such that

$$\begin{aligned}&\nu ^v\left( x^{p+K(x)}\right) =\nu ^v(x^p),\quad i^v\left( x^{p+K(x)}\right) -i^v(x^p)\in 2\mathbf{Z}, \quad \forall p\in \mathbf{N}, \end{aligned}$$

(5.18)

$$\begin{aligned}&k_l\left( x^{p+K(x)}\right) =k_l(x^p), \quad \forall p\in \mathbf{N},\;l\in \mathbf{Z}. \end{aligned}$$

(5.19)

We call K(x) the minimal period of critical modules of iterations of the functional $F_{a, K}$ at x.

For every closed characteristic $(\tau , y)$ on $\Sigma$, let $aT>\tau$ and choose ${\varphi }_a$ as above. Determine $\rho$ uniquely by $\frac{{\varphi }_a'(\rho )}{\rho }=\frac{\tau }{aT}$. Let $x=\rho y(\frac{\tau t}{T})$. Then we define the index $i(\tau , y)$ and nullity $\nu (\tau , y)$ of $(\tau , y)$ by

$$\begin{aligned} i(\tau , y)=i^v(x), \quad \nu (\tau , y)=\nu ^v(x). \end{aligned}$$

Then the mean index of $(\tau , y)$ is defined by

$$\begin{aligned} \hat{i}(\tau , y) = \lim _{m\rightarrow \infty }\frac{i(m\tau , y)}{m}. \end{aligned}$$

(5.20)

Note that by Proposition 2.11 of [22], the index and nullity are well defined and are independent of the choice of a.

For a closed characteristic $(\tau , y)$ on $\Sigma$, we simply denote by $y^m\equiv (m\tau , y)$ the mth iteration of y for $m\in \mathbf{N}$. By Proposition 3.2 of [22], we can define the critical type numbers $k_l(y^m)$ of $y^m$ to be $k_l(x^m)$, where $x^m$ is the critical point of $F_{a, K}$ corresponding to $y^m$. We also define $K(y)=K(x)$.

Remark 5.5

(cf. Remark 4.10 of [22]) Note that $k_l(y^m)=0$ for $l\notin [0, \nu (y^m)-1]$ and it can take only values 0 or 1 when $l=0$ or $l=\nu (y^m)-1$. Moreover, the following facts are useful:

• (i) $k_0(y^m)=1$ implies $k_l(y^m)=0$ for $1\le l\le \nu (y^m)-1$.

• (ii) $k_{\nu (y^m)-1}(y^m)=1$ implies $k_l(y^m)=0$ for $0\le l\le \nu (y^m)-2$.

• (iii) $k_l(y^m)\ge 1$ for some $1\le l\le \nu (y^m)-2$ implies $k_0(y^m)=k_{\nu (y^m)-1}(y^m)=0$.

• (iv) In particular, only one of the $k_l(y^m)\hbox {s}$ for $0\le l\le \nu (y^m)-1$ can be non-zero when $\nu (y^m)\le 3$.

For a closed characteristic $(\tau ,y)$ on $\Sigma$, the average Euler characteristic $\hat{\chi }(y)$ of y is defined by

$$\begin{aligned} \hat{\chi }(y)=\frac{1}{K(y)}\sum _{\begin{array}{c} 1\le m\le K(y)\\ 0\le l\le 2n-2 \end{array}} (-1)^{i(y^{m})+l}k_l(y^{m}). \end{aligned}$$

(5.21)

$\hat{\chi }(y)$ is a rational number. In particular, if all $y^m\hbox {s}$ are non-degenerate, then by Proposition 5.4 we have

$$\begin{aligned} \hat{\chi }(y) = \left\{ \begin{array}{ll} (-1)^{i(y)}, &{}\quad {\mathrm{if}}\quad i(y^2)-i(y)\in 2\mathbf{Z}, \\ \frac{(-1)^{i(y)}}{2}, &{}\quad {\mathrm{otherwise}}. \\ \end{array}\right. \end{aligned}$$

(5.22)

We have the following mean index identities for closed characteristics.

Theorem 5.6

Suppose that ${\Sigma }\in \mathcal{H}_{st}(2n)$ satisfies $\,^{\#}\mathcal{T}({\Sigma })<+\infty$. Denote all the geometrically distinct prime closed characteristics by $\{(\tau _j,\; y_j)\}_{1\le j\le k}$. Then the following identities hold

$$\begin{aligned} \sum _{\begin{array}{c} 1\le j\le k\\ \hat{i}(y_j)>0 \end{array}}\frac{\hat{\chi }(y_j)}{\hat{i}(y_j)}= & {} \frac{1}{2}, \end{aligned}$$

(5.23)

$$\begin{aligned} \sum _{\begin{array}{c} 1\le j\le k\\ \hat{i}(y_j)<0 \end{array}}\frac{\hat{\chi }(y_j)}{\hat{i}(y_j)}= & {} 0. \end{aligned}$$

(5.24)

Let $F_{a, K}$ be a functional defined by (5.4) for some $a, K\in \mathbf{R}$ sufficiently large and let $\epsilon >0$ be small enough such that $[-\epsilon , 0)$ contains no critical values of $F_{a, K}$. For b large enough, The normalized Morse series of $F_{a, K}$ in $X^{-{\epsilon }}{\setminus } X^{-b}$ is defined, as usual, by

$$\begin{aligned} M_a(t)=\sum _{q\ge 0,\;1\le j\le p} \dim C_{S^1,\;q}(F_{a, K}, \;S^1\cdot v_j)t^{q-d(K)}, \end{aligned}$$

(5.25)

where we denote by $\{S^1\cdot v_1, \ldots , S^1\cdot v_p\}$ the critical orbits of $F_{a, K}$ with critical values less than $-\epsilon$. The Poincaré series of $H_{S^1, *}( X, X^{-{\epsilon }})$ is $t^{d(K)}Q_a(t)$, according to Theorem 5.1 of [22], if we set $Q_a(t)=\sum _{k\in \mathbf{Z}}{q_kt^k}$, then

(5.26)

where I is an interval of $\mathbf{Z}$ such that $I \cap [i(\tau , y), i(\tau , y)+\nu (\tau , y)-1]=\emptyset$ for all closed characteristics $(\tau , y)$ on $\Sigma$ with $\tau \ge aT$. Then by Section 6 of [22], we have

$$\begin{aligned} M_a(t)-\frac{1}{1-t^2}+Q_a(t) = (1+t)U_a(t), \end{aligned}$$

(5.27)

where $U_a(t)=\sum _{i\in \mathbf{Z}}{u_it^i}$ is a Laurent series with nonnegative coefficients. If there is no closed characteristic with $\hat{i}=0$, then

$$\begin{aligned} M(t)-\frac{1}{1-t^2}=(1+t)U(t), \end{aligned}$$

(5.28)

where $M(t)=\sum _{i\in \mathbf{Z}}{m_it^i}$ denotes the limit of $M_a(t)$ as a tends to infinity, $U(t)=\sum _{i\in \mathbf{Z}}{u_it^i}$ denotes the limit of $U_a(t)$ as a tends to infinity and possesses only non-negative coefficients. Specially, suppose that there exists an integer $p<0$ such that the coefficients of M(t) satisfy $m_p>0$ and $m_q=0$ for all integers $q<p$. Then (5.28) implies

$$\begin{aligned} m_{p+1} \ge m_p. \end{aligned}$$

(5.29)