论文标题
当地共同扁平的4个manifolds上的共形谐波地图的独特性
Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds
论文作者
论文摘要
引入了两个riemannian表面上的谐波图理论的动机，引入了两个riemannian歧管$ m $和$ n $之间的共形荷尔马人地图，以寻找在一般尺寸的riemannian歧管$ m $上定义的谐波概念。当目标是真实或复数的集合时，它们是共形不变能量功能的关键点，并重新组装GJMS运算符。在四个维歧管上，共形谐波图是内在双谐波映射的共形不变的对应物，也是合成不变的paneitz操作员的映射版本。 在本文中，我们考虑了从某些本地共同扁平的4个manifolds到球体中的共形谐波图。我们证明了这种保形谐波图的定量唯一性结果，这是共形不变的能量功能的直接结果。为此，我们被指导证明了对流形的二阶二阶不平等现象的版本，这可能引起了独立的兴趣。
Motivated by the theory of harmonic maps on Riemannian surfaces, conformal-harmonic maps between two Riemannian manifolds $M$ and $N$ were introduced in search of a natural notion of harmonicity for maps defined on a general even dimensional Riemannian manifold $M$. They are critical points of a conformally invariant energy functional and reassemble the GJMS operators when the target is the set of real or complex numbers. On a four dimensional manifold, conformal-harmonic maps are the conformally invariant counterparts of the intrinsic bi-harmonic maps and a mapping version of the conformally invariant Paneitz operator for functions. In this paper, we consider conformal-harmonic maps from certain locally conformally flat 4-manifolds into spheres. We prove a quantitative uniqueness result for such conformal-harmonic maps as an immediate consequence of convexity for the conformally-invariant energy functional. To this end, we are led to prove a version of second order Hardy inequality on manifolds, which may be of independent interest.
