论文标题
在封闭表面上的涡流型哈密顿量的平衡
Equilibria of vortex type Hamiltonians on closed surfaces
论文作者
论文摘要
我们证明了漩涡类型哈密顿人的关键点的存在 \ [ h（p_1，\ ldots，p_n） = \ sum _ {{{i，j = 1}，{i \ ne j}}}^nγ_Iγ_jg（p_i，p_i，p_j）+ψ（p_1，\ dots，p_n） \]在封闭的riemannian表面$（σ，g）$上，对球体或投射平面不是同件。 $ g $在这里表示Laplace-Beltrami操作员的绿色功能，$σ$，$ψ：σ^n \ to \ mathbb {r} $可能是类$ c^1 $和$ c^1 $的任何功能，$ c^1 $和$γ_1，\ \ \ dots，γ_n\ in \ in \ mathbb {r} r} $ setminus} $ nes vort \ nes vort。来自流体动力学的Kirchhoff-Routh Hamiltonian对应于$ψ= - \ sum_ {i = 1}^nγ_i^2H（p_i，p_i，p_i）$其中$ h：σ\timesς\ to \ mathbb {r} $是laplace-beltrami operatortortortortortortorrator的laplace-beltrami operator。我们获得了关键点$ p =（p_1，\ dots，p_n）$ for nutary $ n $和涡流$（γ_1，\ dots，γ_n）$ in $ \ mathbb {r}^n \ setminus v $ were $ v $在其中明显地给出了Algebraic verthers compimension a coundimimensial consimimensial s Partlicity of Algebraic v $。
We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N Γ_iΓ_jG(p_i,p_j)+ψ(p_1,\dots,p_N) \] on a closed Riemannian surface $(Σ,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $Σ$, $ψ:Σ^N\to\mathbb{R}$ may be any function of class $C^1$, and $Γ_1,\dots,Γ_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $ψ= -\sum_{i=1}^N Γ_i^2h(p_i,p_i)$ where $h:Σ\timesΣ\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(Γ_1,\dots,Γ_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.
