关于边界上存在无处消失静态势的静态流形
On static manifolds with boundary admitting a nowhere-vanishing static potential
- National Research University Higher School of Economics（国立研究高等经济学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究具有边界且存在无处消失静态势的完备静态流形，证明在自然下界条件下其必为正标量曲率、负边界平均曲率且紧致，并给出体积估计、分裂刚性及尖锐平均曲率界等结果。
AI中文摘要：
我们研究具有边界且允许存在无处消失静态势的完备静态流形。我们的主要结果表明，在涉及标量曲率与边界平均曲率的自然下界条件下，具有边界的简单静态流形实际上必须具有正标量曲率、负边界平均曲率，并且是紧致的；我们还获得了涉及流形体积及其边界几何的显式关系与估计。在标量平坦情形下，我们证明了全局分裂与里奇平坦刚性结果，包括边界不连通的情况；而在负标量曲率情形下，我们建立了尖锐的平均曲率界，并通过指数型扭曲积结构刻画了等号情形。证明本质上依赖于对相关爱因斯坦流形的研究。在附录中，我们推导了具有边界的静态流形的若干恒等式，并讨论了无边界情形下的相关爱因斯坦流形技术。
英文摘要：
We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.
