Viktor L. Ginzburg, Basak Z. Gurel
We give a detailed construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.
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We outline the construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.
We construct a new aperiodic symplectic plug and hence new smooth counterexamples to the Hamiltonian Seifert conjecture in R^{2n} for n>2. In other words, we develop an alternative procedure, to those of V. L. Ginzburg and M. Herman, for constructing smooth Hamiltonian flows, on the standard symplectic R^{2n} for n>2, which have compact regular level sets that contain no periodic orbits. The plug described here is a modification of those built by Ginzburg. In particular, we utilize a different "trap" which makes the necessary embeddings of this plug much easier to construct.
The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of counterexamples to the Hamiltonian Seifert conjecture in dimensions greater than or equal to six is outlined, mainly following the method introduced by the author of the paper. The Hamiltonian version of Seifert's theorem is also stated. A list of all known at this moment examples an…
A smooth counterexample to the Hamiltonian Seifert conjecture for six-dimensional symplectic manifolds is found. In particular, we construct a smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4, such that one of its non-singular level sets carries no periodic orbits of the Hamiltonian flow. The function can be taken to be C^0-close and isotopic to a positive-definite quadratic form so that the level set in question is isotopic to an ellipsoid. This is a refinement of previously known constructions giving such functions for 2n > 6. The proof is based on a new version o…
Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits. Very little is known about how large the set of regular energy values without periodic orbits can be. For instance, in all known examples of Hamiltonian flows on linear spaces such energy values form a discrete set, whereas ``almost existence theorems'' only require this set to…
We give a purely geometrical smooth characterization of closed infrasolv manifolds and orbifolds by showing that, up to diffeomorphism, these are precisely the spaces which admit a collapse with bounded curvature and diameter to compact flat orbifolds. Moreover, we distinguish irreducible smooth fake tori geometrically from standard ones by proving that the former have non-vanishing D-minimal volume.
We study totally geodesic codimension 1 smooth foliations on Lorentzian manifold. We are in particular interested by the relations between riemannian flows and geodesic foliations. We prove that, up to a 2-cover, any Seifert bundle admit such a foliation.
A counterexample to the singular Weinstein conjecture
本文在 3 维 b-接触流形上构造 Reeb 向量场，证明奇异周期轨道数目可预先指定，并由此给出文献[23]中奇异 Weinstein 猜想与 Hamilton Seifert 猜想的反例，同时指出广义 Weinstein 猜想仍待解决。
Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $Φ\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $Φ$ is an explicit Seifert hypersurface for $M$.
A Smooth Counterexample to the Trautman Conjecture
本文通过修改不可嵌入的光滑强伪凸CR 3-流形的标准构造，得到特劳特曼猜想的一个强伪凸反例，证明了该猜想不成立。
We construct a sequence of smooth minimizing surfaces in a sequence of metrics converging to the standard Euclidean metric, so that they have diverging $L^2$ norm of second fundamental form.
Counterexamples to the Lorentzian Calderón problem
研究显示非等距的光滑全球超曲面洛伦兹度量在无限圆柱体上具有相同的超几何狄利克雷到神经元映射
We construct smooth metrics on 2-manifold with nonpositive Gauss curvature which cannot be (C^3) locally isometrically embedded in R^3. Moreover, the Gauss curvature of the metric can be made negative except for one point.
In this paper, we present smooth examples of degenerate hyperbolic and mixed type Monge-Ampere equations in the plane, which do not admit a local C^3 solution.
Counterexamples to prescribed scalar curvature in the null case
该研究在$n\boldsymbol{\text{≥}}5$的平坦环面上构造了变号且均值为负的光滑函数，证明其无法作为给定共形类中度量的纯量曲率，给出了零情形下规定纯量曲率的反例。
Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture
本文构造了新的均曲率流扩张器，这些扩张器光滑地趋于某些收缩器产生的锥体。对于每个锥体，证明了存在任意高亏格的扩张器，反驳了伊尔曼恩的亏格减少猜想。
Counterexamples to the non-simply connected Double Soul Conjecture
研究非单连通双核心猜想，通过构造了反例，无穷多个3维反例，以及维度无界增长的平坦反例，我。
A counterexample to Nirenberg's conjecture on closed asymptotic curves
本文构造了R³中一条含两条闭渐近曲线的光滑嵌入环带，为Nirenberg1962年的猜想提供了反例，其构造基于Lelieuvre公式与Riccati方程周期解。
Any closed orientable and smooth non-positively curved manifold M is known to admit a geometric characteristic splitting, analogous to the JSJ decomposition in three dimensions. We show that when this splitting consists of pieces which are Seifert fibered or pieces each of whose fundamental group has non-trivial centre, M collapses with bounded curvature and has zero Perelman invariant.
A counterexample to the bounded mass property
该研究针对紧复流形的有界质量性质，通过证明Hopf三重曲面不具备该性质，解答了Boucksom–Guedj–Lu提出的相关问题。
A contractible Schiffer counterexample on the half-sphere
本文构造了球面上非平凡光滑可缩区域，其 Neumann 特征函数在边界上为常数，通过局部分歧和计算机辅助验证，与 Serrin 型刚性结果形成对比。
Stable ample vector bundles without semipositive Kobayashi curvature
本文在Hirzebruch曲面上构造了无半正Kobayashi曲率的稳定丰沛向量丛，否定了Kobayashi问题，并为Fulton问题和Hartshorne--Schneider猜想提供了反例。
In this letter a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely---taking into account that the conjecture lacks any precise formulation yet---first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically stronger concept of a "robust counterexample". Then making use of Penrose' nonlinear graviton construction (i.e., twistor theory) and a Wick rotation trick we construct a smooth Ricci-flat but not flat Lorentzian metric on the largest member of the Gompf--Taubes uncountable radial …
Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation (after Chen and Wu)
针对常标量曲率与常边界平均曲率方程，本文将低能量解集失去紧性的光滑反例维数阈值由 Chen 和 Wu 的不小于62维改进至不小于35维。
Integral of Scalar Curvature under Volume Collapse
本文构造了三维完备正里奇曲率度量使数量曲率积分发散，否定了丘成桐猜想，并给出紧流形上Naber猜想的反例。
A counterexample is given for the Knaster-like conjecture of Makeev for functions on $S^2$. Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively.
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