Existence of closed characteristics on compact convex hypersurfaces in \(\mathbf{R}^{2n}\)

In this paper, we prove there exist at least \([\frac{n+1}{2}]+1\) geometrically distinct closed characteristics on every compact convex hypersurface \({\Sigma }\) in \(\mathbf{R}^{2n}\), where \(n\ge 2\). In particular, this gives a new proof in the case \(n=3\) to a long standing conjecture in Hamiltonian analysis. Moreover, there exist at least \([\frac{n}{2}]+1\) geometrically distinct non-hyperbolic closed characteristics on \({\Sigma }\) provided the number of geometrically distinct closed characteristics on \({\Sigma }\) is finite.

I would like to sincerely thank the referees for their careful readings of this paper and for their valuable comments and suggestions. I would like to sincerely thank my Ph.D. thesis advisor, Professor Yiming Long, for introducing me to Hamiltonian dynamics and for his valuable help and encouragement during my research. I would like to say how enjoyable it is to work with him.

Authors and Affiliations

1. Key Laboratory of Pure and Applied Mathematics, School of Mathematical Science, Peking University, Beijing, 100871, People’s Republic of China

   Wei Wang

Corresponding author

Correspondence to Wei Wang.

Communicated by P. Rabinowitz.

Partially supported by National Natural Science Foundation of China Nos. 11222105, 11431001, Foundation for the Author of National Excellent Doctoral Dissertation of People’s Republic of China No. 201017.

Appendix: Index iteration theory for closed characteristics

In this section, we review briefly the index theory for symplectic paths developed by Long and his coworkers. All the details can be found in [16].

The symplectic group \(\mathrm{Sp}(2n)\) is defined by

For \(\tau >0\) we consider paths in \(\mathrm{Sp}(2n)\):

As in [15] we define

For \({\omega }\in \mathbf{U}\), we define the following codimension 1 hypersurface in \(\mathrm{Sp}(2n)\) as in [15]:

For any \(M\in \mathrm{Sp}(2n)_{{\omega }}^0\), we define the co-orientation of \(\mathrm{Sp}(2n)_{{\omega }}^0\) at M by the positive direction \(\frac{d}{dt}Me^{t J}|_{t=0}\) of the path \(Me^{t J}\) with \(0\le t\le {\epsilon }\) and \({\epsilon }>0\) being small enough. Denote by

For any two paths \(\xi \) and \(\eta :[0,\tau ]\rightarrow \mathrm{Sp}(2n)\) with \(\xi (\tau )=\eta (0)\), we define as usual:

As in [16], for any two \(2m_k\times 2m_k\) matrices of square block form \(M_k=\left( \begin{array}{ll}A_k&{}B_k\\ C_k&{}D_k\\ \end{array}\right) \) with \(k=1, 2\), we define the \(\;\mathrm{\diamond }\)-product of \(M_1\) and \(M_2\) to be the following \(2(m_1+m_2)\times 2(m_1+m_2)\) matrix \(M_1\mathrm{\diamond }M_2\):

Denote by \(M^{\mathrm{\diamond }k}\) the k-fold \(\mathrm{\diamond }\)-product \(M\mathrm{\diamond }\cdots \mathrm{\diamond }M\). One can easily check that the \(\mathrm{\diamond }\)-product of any two symplectic matrices is symplectic. For any two paths \({\gamma }_j\in \mathcal{P}_{\tau }(2n_j)\) with \(j=0\) and 1, let \({\gamma }_0\mathrm{\diamond }{\gamma }_1(t)= {\gamma }_0(t)\mathrm{\diamond }{\gamma }_1(t)\) for \(t\in [0,\tau ]\).

We define a special path \(\xi _n\):

Definition 4.1

(cf. [15, 16]) For \({\omega }\in \mathbf{U}\) and \(M\in \mathrm{Sp}(2n)\), define

For \(\tau >0\) and \({\gamma }\in \mathcal{P}_{\tau }(2n)\), define

If \({\gamma }\in \mathcal{P}_{\tau ,{\omega }}^{*}(2n)\), define

where the right hand side of (4.4) is the usual homotopy intersection number, and the orientation of \({\gamma }*\xi _n\) is its positive time direction under homptopy with fixed end points.

If \({\gamma }\in \mathcal{P}_{\tau ,{\omega }}^0(2n)\), we let \(\mathcal {F}({\gamma })\) be the set of all open neighborhoods of \({\gamma }\) in \(\mathcal{P}_{\tau }(2n)\), and define

Then the tuple

is called the index function of \({\gamma }\) at \({\omega }\).

For a symplectic path \({\gamma }\in \mathcal{P}_{\tau }(2n)\) and \(m\in \mathbf{N}\), we define its mth iteration \({\gamma }^m:[0,m\tau ]\rightarrow \mathrm{Sp}(2n)\) to be

We also denote the extended path on \([0,+\infty )\) by \({\gamma }\).

Definition 4.2

(cf. [15, 16]) For \({\gamma }\in \mathcal{P}_{\tau }(2n)\), define

We define the mean index \(\hat{i}({\gamma },m)\) per \(m\tau \) for \(m\in \mathbf{N}\) to be

For \(M\in \mathrm{Sp}(2n)\) and \({\omega }\in \mathbf{U}\), we define the splitting numbers \(S_M^{\pm }({\omega })\) of M at \({\omega }\) to be

where \({\gamma }\in \mathcal{P}_{\tau }(2n)\) is any path satisfying \({\gamma }(\tau )=M\).

Given a path \(\gamma \in \mathcal{P}_{\tau }(2n)\), we want to deform it to a new path \(\eta \) in \(\mathcal{P}_{\tau }(2n)\) such that

and that \((i_1(\eta ^m),\nu _1(\eta ^m))\) is easy enough to compute. This leads to finding homotopies \(\delta :[0,1]\times [0,\tau ]\rightarrow \mathrm{Sp}(2n)\) starting from \(\gamma \) in \(\mathcal{P}_{\tau }(2n)\) and keeping the end points of the homotopy always stay in a certain suitably chosen maximal subset of \(\mathrm{Sp}(2n)\) so that (4.10) holds. Actually this set was first discovered in [15] as the path connected component \(\Omega ^0(M)\) containing \(M=\gamma (\tau )\) of the set

Here \(\Omega ^0(M)\) is called the homotopy component of M in \(\mathrm{Sp}(2n)\).

Note that we have the following \(2\times 2\) and \(4\times 4\) symplectic matrix as basic normal forms (cf. [15, 16]):

where \(b=\left( \begin{array}{llll}b_1 &{} b_2\\ b_3 &{} b_4\\ \end{array}\right) \) with \(b_i\in \mathbf{R}\) and \(b_2\not =b_3\). Moreover, we call \( N_2(\omega , b)\) trivial if \((b_2-b_3)\sin \theta >0\); and non-trivial if \((b_2-b_3)\sin \theta <0\).

We have the following properties for Splitting numbers:

Lemma 4.3

(cf. [15] and Lemma 9.1.5 of [16]) Splitting numbers \(S_M^{\pm }({\omega })\) are independent of the choice of the path \({\gamma }\in \mathcal{P}_\tau (2n)\) satisfying \({\gamma }(\tau )=M\) appeared in (4.9). Moreover, for \({\omega }\in \mathbf{U}\) and \(M\in \mathrm{Sp}(2n)\), splitting numbers \(S_N^{\pm }({\omega })\) are constant for all \(N\in {\Omega }^0(M)\).

Lemma 4.4

(cf. [15], Lemma 9.1.5 and List 9.1.12 of [16]) For \(M\in \mathrm{Sp}(2n)\) and \({\omega }\in \mathbf{U}\), we have

For \(M_i\in \mathrm{Sp}(2n_i)\) with \(i=0\) and 1, there holds

We have the following symplectic additivity property for index functions:

Theorem 4.5

(cf. Theorem 6.1 of [17] or Theorem 6.2.7 of [16]) For any \(\gamma _j\in \mathcal{P}_\tau (2n_j)\) with \(j=0, 1\), we have

Now we use the above defined index function to study the Morse indices of the Clarke–Ekeland dual functional \(\Phi \) at a critical point. Let \(\Sigma \in \mathcal{H}(2n)\). Using notations in §1, for any closed characteristic \((\tau ,y)\) and \(m\in \mathbf{N}\), we define its mth iteration \(y^m:\mathbf{R}/(m\tau \mathbf{Z})\rightarrow \mathbf{R}^{2n}\) by

We still denote by y its extension to \([0,+\infty )\).

We define via Definition 4.2 the following

for all \(m\in \mathbf{N}\), where \({\gamma }_y\) is the associated symplectic path of \((\tau ,y)\).

The following theorem describe the relation between the above defined indices and the indices defined by Ekeland [2–4]. Thus we can compute the Morse indices of \(\Phi \) at a critical point by those of its associated symplectic paths.

Theorem 4.6

(cf. Lemma 1.1 of [17], Theorem 15.1.1 of [16]) Suppose \((\tau ,y)\) is a closed characteristic on \({\Sigma }\). Then we have

where \(i(y^m)\) and \(\nu (y^m)\) are the index and nullity defined by Ekeland [2–4]. In particular, we have \(\hat{i}(y)=\hat{i}(y,\, 1)\), where \(\hat{i}(y)\) is the mean index of \((\tau ,\,y)\) defined by Ekeland.

The following is the precise index iteration formulae for symplectic paths, which is due to Long (cf. Theorem 8.3.1 and Corollary 8.3.2 of [16]). Using this formula, we can compute the indices of any iteration of a fixed symplectic path whenever we know its initial index and end point in \(\mathrm{Sp}(2n)\).

Theorem 4.7

Let \(\gamma \in \mathcal{P}_\tau (2n)\), Then there exists a path \(f\in C([0,1],\Omega ^0(\gamma (\tau ))\) such that \(f(0)=\gamma (\tau )\) and

where \( N_2(\omega _j, u_j) \)s are non-trivial and \( N_2({\lambda }_j, v_j)\)s are trivial basic normal forms; \(\sigma (M_0)\cap U=\emptyset \); \(p_-\), \(p_0\), \(p_+\), \(q_-\), \(q_0\), \(q_+\), r, \(r_*\) and \(r_0\) are non-negative integers; \(\omega _j=e^{\sqrt{-1}\alpha _j}\), \( \lambda _j=e^{\sqrt{-1}\beta _j}\); \(\theta _j\), \(\alpha _j\), \(\beta _j\) \(\in (0, \pi )\cup (\pi , 2\pi )\); these integers and real numbers are uniquely determined by \(\gamma (\tau )\). Then using the functions defined in (1.7).

Moreover, we have \(i(\gamma , 1)\) is odd if \(f(1)=N_1(1, 1)\), \(I_2\), \(N_1(-1, 1)\), \(-I_2\), \(N_1(-1, -1)\) and \(R(\theta )\); \(i(\gamma , 1)\) is even if \(f(1)=N_1(1, -1)\) and \( N_2(\omega , b)\); \(i(\gamma , 1)\) can be any integer if \(\sigma (f(1)) \cap \mathbf{U}=\emptyset \).

We have the following iteration inequalities for the index function:

Theorem 4.8

(cf. Theorem 2.3 of [17]) Let \(\gamma \in \mathcal{P}_\tau (2n)\) and \(M = \gamma (\tau )\). Suppose that there exist \(P\in \mathrm{Sp}(2n)\) and \(Q \in \mathrm{Sp}(2n -2)\) such that \(M=P^{-1}(N_1(1, 1)\diamond Q)P\). Then for any \(m\in \mathbf{N}\), there holds

where e(M) is the total algebraic multiplicity of all eigenvalues of M on the unit circle in the complex plane \(\mathbf{C}\).

The following is the common index jump theorem of Long and Zhu.

Theorem 4.9

(cf. Theorems 4.1–4.4 of [17]) Let \(\gamma _k\in \mathcal{P}_{\tau _k}(2n)\) for \( k = 1,\ldots ,q\) be a finite collection of symplectic paths. Let \(M_k = \gamma _k(\tau _k)\). Suppose that there exist \(P_k\in \mathrm{Sp}(2n)\) and \(Q _k\in \mathrm{Sp}(2n -2)\) such that \(M_k=P_k^{-1}(N_1(1, 1)\diamond Q_k)P_k\) and \(\hat{i}(\gamma _k, 1) > 0\), for all \(k = 1,\ldots ,q\). Then there exist infinitely many \((T,m_1,\ldots ,m_q)\in \mathbf{N}^{q+1}\) such that

for every \(k=1,\ldots ,q\). Moreover we have

whenever \(e^{\sqrt{-1}\theta }\in \sigma (M_k)\) and \(\delta \) can be chosen as small as we want (cf. (4.43) of [17]). More precisely, by (4.10) and (4.40) in [17], we have

where \(\chi _k=0\) or 1 for \(1\le k\le q\) and \(\frac{M\theta }{\pi }\in \mathbf{Z}\) whenever \(e^{\sqrt{-1}\theta }\in \sigma (M_k)\) and \(\frac{\theta }{\pi }\in \mathbf{Q}\) for some \(1\le k\le q\). Furthermore, given \(M_0\in \mathbf{N}\), by the proof of Theorem 4.1 of [17], we may further require \(M_0|T\) (since the closure of the set \(\{\{Tv\}:T\in \mathbf{N}, \;M_0|T\}\) is still a closed additive subgroup of \(\mathbf T^h\) for some \(h\in \mathbf{N}\), where we use notations as (4.21) in [17]. Then we can use the proof of Step 2 in Theorem 4.1 of [17] to get T).

In fact, Let \(\mu _i=\sum _{\theta \in (0, 2\pi )}S_{M_i}^-(e^{\sqrt{-1}\theta })\) for \(1\le i\le q\) and \(\alpha _{i, j}=\frac{\theta _j}{\pi }\) where \(e^{\sqrt{-1}\theta _j}\in \sigma (M_i)\) for \(1\le j\le \mu _i\) and \(1\le i\le q\). Let \(h=q+\sum _{1\le i\le q}\mu _i\) and

Then the above theorem is equivalent to finding a vertex

of the unit cube \([0,\,1]^h\) and infinitely many integers \(T\in \mathbf{N}\) such that

for any given \(\epsilon \) small enough (cf. pp. 346 and 349 of [17]).

The next theorem describe how to choose a vertex satisfying (4.38).

Theorem 4.10

(cf. Theorem theo4.2 of [17]) Let H be the closure of the subset \(\{\{mv\}|m\in \mathbf{N}\}\) in \(\mathbf {T}^h=(\mathbf{R}/\mathbf{Z})^h\) and \(V=T_0\pi ^{-1}H\) be the tangent space of \(\pi ^{-1}H\) at the origin in \(\mathbf{R}^h\), where \(\pi :\mathbf{R}^h\rightarrow \mathbf {T}^h\) is the projection map. Define

Define \(\psi (x)=0\) when \(x\ge 0\) and \(\psi (x)=1\) when \(x<0\). Then for any \(a=(a_1,\ldots , a_h)\in A(V)\), the vector

makes (4.38) hold for infinitely many \(T\in \mathbf{N}\).

Moreover, we have the following property for the set A(v):

Theorem 4.11

(cf. Theorem 4.2 of [17])

1. (i)

   If \(v\in \mathbf{R}^h{\setminus }\mathbf{Q}^h\), then \(\dim V\ge 1\), \(0\notin A(v)\subset V\), \(A(v)=-A(v)\) and A(v) is open in V.

2. (ii)

   If \(\dim V = 1\), then \(A(v) = V {\setminus }\{0\}\).

3. (iii)

   If \(\dim V \ge 2\), then A(v) is obtained from V by deleting all the coordinate hyperplanes with dimension strictly smaller than \(\dim V\).

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