论文标题
在表面上的谐波透明扭曲产品
Harmonic-curvature warped products over surfaces
论文作者
论文摘要
对于具有谐波曲率的翘曲产品，非恒定翘曲功能$ ϕ $，以及紧凑的二维基础$（m，h）$，我们建立了二分法：高斯曲率$ k $的度量$ g = ϕ^{-2} h $是恒定和负面的，或$ k $ $ k $ $ k $ k $ k $ k $ k $ k $ k. $ \ varepsilon $的纤维。在这两种情况下，光纤都必须是$ p> 1 $和$ \ varepsilon> 0 $的爱因斯坦歧管，而功能$ f = ϕ^{p/2} $满足yamabe-type的二阶微分方程，on $（m，g）$。我们证明，在大于$ 1 $的属的每个可定向的表面上都实现了这两种可能性，在后一种情况下（也发生在$ 2 $ -SPHERE和真实的投射平面上），所讨论的指标构成了许多独特的同种类型。
For warped products with harmonic curvature, nonconstant warping functions $ϕ$, and compact two-dimensional bases $(M,h)$, we establish a dichotomy: either the Gaussian curvature $K$ of the metric $g=ϕ^{-2}h$ is constant and negative, or $ϕ$ equals a specific elementary function of $K$, also depending on the dimension $p$ and Einstein constant $\varepsilon$ of the fibre. In both cases the fibre must be an Einstein manifold with $p>1$ and $\varepsilon>0$, while the function $f=ϕ^{p/2}$ satisfies a Yamabe-type second-order differential equation on $(M,g)$. We prove that both possibilities are realized on every closed orientable surface of genus greater than $1$, and in the latter case -- which also occurs on the $2$-sphere and real projective plane -- the metrics in question constitute uncountably many distinct homothety types.
