Closed characteristics on non-degenerate star-shaped hypersurfaces in R2n

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula of C. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n ＞ 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics when n=2.     

On a theorem by Ekeland-Hofer

In [EH89, Theorem 1] Ekeland-Hofer prove that for a centrally symmetric, restricted contact type hypersurface in ℝ²ⁿ and for any global, centrally symmetric Hamiltonian perturbation there exists a leaf-wise intersection point. In this note we show that if we replace restricted contact type by star-shaped, there exist infinitely many leaf-wise intersection points or a leaf-wise intersection point on a closed characteristic.     

Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ℝ<Superscript>2<Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface ∑ ⊂ R²ⁿ, there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on ∑ when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on ∑ and at least (n-1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface ∑ ⊂ R²ⁿ index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (τ, y) on ∑ 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.     

Hyperbolic characteristics on star-shaped hypersurfaces

In this paper, we study the stability of closed characteristics on a starshaped compact smooth hypersurface Σ in ²ⁿ. We show that the Maslov-type mean index of such a closed characteristic is independent of the choice of the Hamiltonian functions, and prove that on Σ either there are infinitely many closed characteristics, or there exists at least one nonhyperbolic closed characteristic, provided every closed characteristic possesses its Maslov-type mean index greater than 2 when n is odd, and greater than 1 when n is even.     

The Brake Orbits of Hamiltonian Systems on Positive-type Hypersurfaces

This paper deals with the brake orbits of Hamiltonian system on given energy hypersurfaces Σ = H ⁻¹(1). We introduce a class of contact type but not necessarily star-shaped hypersurfaces in ℝ²ⁿ and call them normalized positive-type hypersurfaces. By using of the critical point theory, we prove that if Σ is a partially symmetric normalized positive-type hypersurface, it must carries a brake orbit of (HS). Furthermore, we obtain some multiplicity results under certain pinching conditions. Our results include the earlier works on this subject given by P. Rabinowitz and A. Szulkin in star-shaped case. An example of partially symmetric normalized positive-type hypersurface in ℝ⁴ that is not star-shaped is also presented Partially supported by NNSF of China (10571085) and Science Foundation of Hohai University.     

The fixed energy problem for a class of nonconvex singular Hamiltonian systems

We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of “strong force” type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type.     

The Stability of Subharmonic Solutions for Hamiltonian Systems

In this paper, by using the dual Morse index theory, we study the stability of subharmonic solutions of the non-autonomous Hamiltonian systems. We obtain a (infinite) sequence of geometrically distinct periodic solutions such that every element has at most one direction of instability (i.e., it has at least 2n − 2 Floquet multipliers lying on the unit circle in the complex plane if the periodic solution is non-degenerate) or it is elliptic (all its 2n Floquet multipliers are lying on the unit circle) if the periodic solution is degenerate.     

Orthogonal separable Hamiltonian systems on <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript>

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T ² for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T ² can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way. This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10671123, 10231020), “Dawn” Program of Shanghai Education Comission of China (Grant No. 03SG10) and Program for New Century Excellent Tatents in University of China (Grant No. 050391)     

本刊英文版Vol.34(2018),No.1论文摘要（英文）

正Non-hyperbolic Closed Characteristics on Non-degenerate Star-shaped Hypersurfaces in R~(2n)Hua Gui DUAN Hui LIU Yi Ming LONG Wei WANGAbstract In this paper,we prove that for every index perfect non-degenerate compact starshaped hypersurface∑(?)R~(2n),there exist at least n non-hyperbolic closed characteristics with even Maslov-type indices on E when n is even.When n is odd,there exist at least n closed     

On a CR Transversality Problem Through the Approach of the Chern–Moser Theory

In this paper, we give a geometric condition for a CR map, which sends a CR non-umbilical Levi non-degenerate hypersurface in ? ⁿ⁺¹ into the hyperquadric in ? ⁿ⁺² with the same signature, to be CR transversal.     

Seifert Conjecture in the Even Convex Case

In this paper, we prove that there exist at least n geometrically distinct brake orbits on every C² compact convex symmetric hypersurface Σ in ?²ⁿ satisfying the reversible condition NΣ = Σ with N = diag(?I_n,Iⁿ). As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integern. © 2014 Wiley Periodicals, Inc.     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

Polyvector representations of GL<Subscript>n</Subscript>

In the present paper, we characterize ⋀ⁿ(GL(n, R)) over any commutative ring R as the connected component of the stabilizer of the Plücker ideal. This folk theorem is classically known for algebraically closed fields and should also be well known in general. However, we are not aware of any obvious reference, so we produce a detailed proof, which follows a general scheme developed by W.C.Waterhouse. The present paper is a technical preliminary to a subsequent paper, where we construct the decomposition of transvections in polyvector representations of GL _n. Bibliography: 50 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 69–97.     

Isoperimetric bounds for the first eigenvalue of the Laplacian

In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space R ⁿ with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an embedded closed hypersurface in R ⁿ .     

次二次Hamilton系统的对偶Morse指标及其闭特征的稳定性

本文通过Ｇｅｌｅｒｋｉｎ逼近方法,在没有任何凸的条件下,研究了次二次Ｈａｍｉｌ－ｔｏｎ系统的ｋ对偶Ｍｏｒｓｅ指标理论．作为应用,在本文研究了Ｒ２ｎ中的凸超曲面上的闭特征的稳定性,证明了在一个较宽松的夹条件下,这类超曲面上至少有一条椭圆闭特征．     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-harmonic forms and stable hypersurfaces in space forms

We prove that a complete noncompact orientable stable minimal hypersurface in \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} or \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}}.     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

Existence d'orbites périodiques complètement elliptiques des Hamiltoniens convexes présentant certaines symétries

Using certain quadratic forms associated to symplectic endomorphisms which we compare with the Clarke-Ekeland dual action functional, we prove: THEOREM. — Let H be a C²-Hamiltonian defined on R²ⁿ, strictly convex, proper and invariant under a certain symplectic rational positive and non-degenerate rotation (this is defined in the introduction); then, every hypersurface of H contains a completely elliptic periodic orbit. This generalizes the result of G. Dell'Antonio, B. D'Onofrio and I. Ekeland contained in [1].     

Closed characteristics on non-compact hypersurfaces in $${\mathbb {R}^{2n}}$$

Viterbo demonstrated that any (2n − 1)-dimensional compact hypersurface of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (, ω). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H _i (M), i = n, . . . , 2n − 1, and the existence of closed characteristics in the non-compact case (including the compact case). J. B. van den Berg is supported by NWO VENI grant 639.031.204. R. C. Vandervorst and F. Pasquotto are supported by NWO VIDI grant 639.032.202. This research is also partially supported by the RTN project ‘Fronts-Singularities’.