论文标题
正质量定理和圆环刚度定理的稳定性在整体曲率界限下
Stability of the positive mass theorem and torus rigidity theorems under integral curvature bounds
论文作者
论文摘要
D. Stern和Bray-Kazaras-Khuri-Sern的作品提供了差异几何身份，这些几何身份将Riemannian 3-manifolds的标态曲率与谐波功能有关。这些定量公式对于稳定性结果很有用，并显示出对这种类型的更多应用的希望。在本文中，我们分析了谐波图到平坦模型空间，以解决有关正质量定理和Geroch猜想的几何稳定性的猜想。通过施加积分的RICCI曲率和等距界限，我们利用了前面提到的公式来对这些谐波图建立强大的控制。当渐近平坦的歧管的质量足够小时，或者当里曼氏圆环几乎具有非阴性标态曲率时，我们将地图升级为差异性，并使定量的Hölder接近到模型空间。
Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in terms of harmonic functions. These quantitative formulas are useful for stability results and show promise for more applications of this type. In this paper, we analyze harmonic maps to flat model spaces in order to address conjectures concerning the geometric stability of the positive mass theorem and the Geroch conjecture. By imposing integral Ricci curvature and isoperimetric bounds, we leverage the previously mentioned formulas to establish strong control on these harmonic maps. When the mass of an asymptotically flat manifold is sufficiently small or when a Riemannian torus has almost non-negative scalar curvature, we upgrade the maps to diffeomorphisms and give quantitative Hölder closeness to the model spaces.
