论文标题
对称beltrami流的存在和结构紧凑$ 3 $ -Manifolds
Existence and structure of symmetric Beltrami flows on compact $3$-manifolds
论文作者
论文摘要
我们表明，对于riemannian歧管的几乎每个给定的对称转换，都存在卷曲算子的特征向量场，对应于遵守对称性的非零特征值。 More precisely, given a smooth, compact, oriented Riemannian $3$-manifold $(\bar{M},g)$ with (possibly empty) boundary and a smooth flow of isometries $ϕ_t:\bar{M}\rightarrow \bar{M}$ we show that, if $\bar{M}$ has non-empty boundary or if the infinitesimal generator is not纯谐波，有一个光滑的向量字段$ x $，与边界有太切线，这是卷曲的特征菲尔德，满足$（ϕ_t）_ {*} x = x $，即在对称转换的推动下是不变的。然后，我们继续证明，如果所涉及的数量是真实的分析数量，并且$（\ bar {m}，g）$具有非空边界，则Arnold的结构定理适用于curl的所有特征菲尔德，遵守对称性和适当的边界条件。更广泛地说，我们表明该结构定理适用于遵守某些非平凡对称性的非变化螺旋性的所有实际分析矢量场。我们证明的副产品是对紧凑，连接，可定向的$ 3 $ manifolds的真实分析杀戮场的表征的表征。
We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian $3$-manifold $(\bar{M},g)$ with (possibly empty) boundary and a smooth flow of isometries $ϕ_t:\bar{M}\rightarrow \bar{M}$ we show that, if $\bar{M}$ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field $X$, tangent to the boundary, which is an eigenfield of curl and satisfies $(ϕ_t)_{*}X=X$, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and $(\bar{M},g)$ has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable $3$-manifolds with and without boundary.
