论文标题
在Riemannian 3-spheres中存在恒定平均曲率2-Spheres
Existence of constant mean curvature 2-spheres in Riemannian 3-spheres
论文作者
论文摘要
我们证明，几乎每个处方的平均曲率几乎都存在分支浸入的浸入式浸入式恒定的平均曲率2-spher中，几乎每个规定的平均曲率，而且对于所有规定的平均曲率，当三个球体呈阳性弯曲时。为了实现这一目标，我们为加权的迪里奇能量功能开发了一个最低最大方案。我们的方法中有三种主要成分：双谐波近似程序，以获得新功能的紧凑性，对Min-Max值的衍生估计值，以获得Min-Max序列的能量上限，几乎每种选择平均曲率选择，以及Morse指数估算以获得另一个均匀的能量，以获得另一个均匀的能量，以达到持续的平均均值稳定曲率，在保持稳定的曲率范围内均具有良好的稳定性。
We prove the existence of branched immersed constant mean curvature 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min-max values to gain energy upper bounds for min-max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.
