The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of F4(-20)
The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of F4(-20)
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F4(-20) 八元接触流形和凸协紧子群上的 Yamabe 算子和不变量
DOI：
10.1007/s10231-021-01093-7
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发表时间：
2021
影响因子：
1
通讯作者：
Wang Wei
中科院分区：
文献类型：
--
作者：
Shi Yun;Wang Wei
一个八元数接触（OC）流形总是球形的。构造了OC流形上的OC Yamabe算子，并证明了其在共形OC变换下的变换公式。一个OC流形是纯量正的、负的或零的当且仅当它的OC Yamabe不变量分别是正的、负的或零。在纯量正OC流形上，我们可以构造OC Yamabe算子的绿色函数，并利用它构造共形不变张量。如果OC正质量猜想成立，它就成为OC度量。我们还证明了两个标量正OC流形的连通和是标量正的，如果颈部是足够长的。在由F4（−20）的凸余紧子群构造的OC流形上，我们构造了一个Nayatani型Carnot-Carathéodory度量。作为推论，这样的OC流形是纯量正的，负的或为零的当且仅当子群的庞加莱临界指数分别小于，大于或等于10。
An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positive OC manifolds to be scalar positive if the neck is sufficiently long. On the OC manifold constructed from a convex cocompact subgroup of F4(−20), we construct a Nayatani-type Carnot–Carathéodory metric. As a corollary, such an OC manifold is scalar positive, negative or vanishing if and only if the Poincaré critical exponent of the subgroup is less than, greater than or equal to 10, respectively.