论文标题
曲率在具有边界的流形上的阳性
Positivity of Curvature on Manifolds with Boundary
论文作者
论文摘要
考虑一个紧凑型歧管$ m $，具有光滑的边界$ \ partial m $。假设$ g $和$ \ tilde {g} $是$ m $上的两个Riemannian指标。我们在$ m $上构建一个指标家庭，该指标与$ \ partial m $的附近的$ g $一致，并同意$ \ tilde {g} $在$ \ partial m $的附近。我们证明，指标家族在边界数据的适当假设下保留了各种自然曲率条件。此外，在对边界数据的合适假设下，我们可以将一个完全大地边界的度量标准变形，同时保留各种自然曲率条件。
Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.
