Radu Slobodeanu
We present new steady Euler solutions on the (round) 3-sphere, that bifurcate from an ansatz proposed by Khesin, Kuksin and Peralta-Salas, showing that these previously known solutions are not isolated. We also extend this ansatz to any Sasakian 3-manifold, such as the Heisenberg group and $SL(2, \mathbb{R})$.
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In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.
In this paper, we characterize the potential function $f$ of the almost conformal gradient Ricci soliton on a Sasakian manifold in terms of the non-dynamical scalar field $p$ and deduce the necessary condition for the potential function $f$ to be constant. Furthermore, a relation between $λ$ and the potential function $f$ has been established. Finally, we prove a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton and also a characterization of the soliton in terms of shrinking, steady or expanding has been done.
We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magnetic relaxation and the hydrodynamic stability of steady Euler flows. We characterize the Sasakian manifolds where the Reeb field is a minimizer in terms of the first positive eigenvalue of the curl operator and show that for $a>a_0$ (a constant that depends on the Sasakian struc…
Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow…
In this paper we provide a local construction of a Sasakian manifold given a Kähler manifold. Obatined in this way manifold we call Sasakian lift of Kähler base. Almost contact metric structure is determined by the operation of the lift of vector fields - idea similar to lifts in Ehresmann connections. We show that Sasakian lift inherits geometry very close to its Kähler base. In some sense geometry of the lift is in analogy with geometry of hypersurface in Kähler manifold. There are obtained structure equations between corresponding Levi-Civita connections, curvatures and Ricci tensors of the…
Using $S^1$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn-Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of $S^2\times S^3$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli spac…
It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n + 1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting…
We point out a duality between steady incompressible Euler flows and solutions of the strongly coupled Faddeev-Skyrme sigma model with potential (mass) term. We supplement this result with various applications and several explicit examples.
A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution
本文构造了三维球面上具有两个独立首次积分且与到二维球面的近浸没纤维相切的稳态欧拉流，并证明该浸没是带标准势的四次Faddeev-Skyrme模型的临界点。
We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we show that the steady Euler flows cannot be constructed using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in higher dimensions are also proved.
A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.
In this article we construct a smooth Euler flow supported in a neighborhood of a helix. It may be considered a generalization of a similar solution found by the author for a circle.
The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a counterexample by Sullivan. However, it is satisfied under the geometric condition of being geodesible. In this work, we use the recent characterization of Eulerisable flows (or more generally flows admitting a strongly adapted one-form) to prove that the conjecture remains true for this larger class of vector fields.
Steady Euler flows and Beltrami fields in high dimensions
本文利用开书分解，在任意奇数维流形上为欧拉方程构造非消失稳态解，实现于接触Reeb场的不变子流形，并刻画贝尔特拉米场，与三维情形对比，构造无周期轨道的非消失贝尔特拉米场。
We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler fl…
This paper answers some questions about conjugate points along the geodesics corresponding to steady 2D Euler flows, posed by a paper of Drivas-Misiolek-Shi-Yoneda. We present a new sufficient criterion for the existence of conjugate points, which improves on the criterion of Misiolek. It applies in any rotational cell of a steady 2D Euler flow, and in case of rotational symmetry it captures all known conjugate points. We give a general construction of the surfaces that admit a steady fluid with given area form, velocity profile, and vorticity profile, and from this we show how to detect conju…
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