论文标题
$（\ Mathbb {s}^2 \ times \ Mathbb {s}^2）\ operatorAname {\ propatorAtorname {\ Mathbb {s Mathbb {s}^2 \ times times^2 \ times \ mathbb {s s}^2）
Conformally formal manifolds and the uniformly quasiregular non-ellipticity of $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$
论文作者
论文摘要
We show that the manifold $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$ does not admit a non-constant non-injective uniformly quasiregular self-map.这回答了Martin，Mayer和Peltonen的一个问题，并提供了一个准椭圆形的歧管歧管的第一个例子，该歧管并非均匀地椭圆形。 为了获得结果，我们引入了形式形式的形式，它们是封闭的平滑$ n $ - manifolds $ m $，承认$（n/k）$ - 谐波$ k $的结构$ [g] $ $ [g] $形式的$（n/k）$ - 谐波$ k $ forms $ [g] $。这与现有的几何形式流形研究相当。我们表明，与几何形式的理论类似，真正的共同体学环$ h^*（m; \ \ m m i \ mathbb {r}）$的正式形式$ n $ -n $ -manifold $ m $承认代数$φ\ colon h^*（m; \ mathbb {r}） \ mathbb {r}^n $。我们还表明，从更强的意义上讲，均匀的椭圆形歧管$ m $是形式上正式的，其中楔形产品被缩放的clifford产品代替。对于这种更强大的子宫形式，$φ$的图像是在$ \ wedge^* \ mathbb {r}^n $的欧几里得Clifford产品下关闭的，而这反过来又不可能$ m =（\ mathbb {s} s}^2 \ times \ times \ times \ times \ mathbb {s}^2） \ times \ mathbb {s}^2）$。
We show that the manifold $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$ does not admit a non-constant non-injective uniformly quasiregular self-map. This answers a question of Martin, Mayer, and Peltonen, and provides the first example of a quasiregularly elliptic manifold which is not uniformly quasiregularly elliptic. To obtain the result, we introduce conformally formal manifolds, which are closed smooth $n$-manifolds $M$ admitting a measurable conformal structure $[g]$ for which the $(n/k)$-harmonic $k$-forms of the structure $[g]$ form an algebra. This is a conformal counterpart to the existing study of geometrically formal manifolds. We show that, similarly as in the geometrically formal theory, the real cohomology ring $H^*(M; \mathbb{R})$ of a conformally formal $n$-manifold $M$ admits an embedding of algebras $Φ\colon H^*(M; \mathbb{R}) \hookrightarrow \wedge^* \mathbb{R}^n$. We also show that uniformly quasiregularly elliptic manifolds $M$ are conformally formal in a stronger sense, in which the wedge product is replaced with a conformally scaled Clifford product. For this stronger version of conformal formality, the image of $Φ$ is closed under the Euclidean Clifford product of $\wedge^* \mathbb{R}^n$, which in turn is impossible for $M = (\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$.
