具有合成Ricci界和单射半径为正的空间的正则性与结构
Regularity and structure of spaces with synthetic Ricci bounds and positive injectivity radius
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中文总结 AI 辅助
本文为具合成Ricci界的RCD空间建立结构理论，在单射半径为正的假设下重构光滑微分结构并证正则性结果，还建立调和半径下界等，证实了V. Kapovitch的相关猜想。
中文摘要 AI 辅助
本文针对具有合成下Ricci曲率界的度量测度空间（即RCD空间）建立了一套结构理论。在单射半径为正的假设下，我们重构出光滑微分结构并证明了正则性结果：这类空间是$W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$型黎曼流形，其权函数也属于$W^{1,p}_{\mathrm{loc}}\cap C^{0,\alpha}_{\mathrm{loc}}$，适用于所有$p<\infty$和$\alpha \in (0,1)$的距离图与调和图。我们还建立了调和半径的定量下界，以及紧性定理和定量几何界。作为副产品，我们在该框架下发展了一套完整的椭圆正则性理论。这些结果尤其适用于光滑加权黎曼流形和满足合成曲率上界（即CBA条件）的RCD空间，即便在这些情形下，所得正则性结论也是新的；在后一情形中，黎曼度量和权函数均被证明是局部Lipschitz的。作为进一步应用，我们建立了纤维化定理，并利用度量平滑化在合成框架下证实了V. Kapovitch关于具有混合曲率界的几乎平坦流形的猜想。
英文摘要
In the paper, we develop a structure theory for metric measure spaces with synthetic lower Ricci curvature bounds, known as RCD spaces. Under a positive injectivity-radius assumption, we recover a smooth differential structure and prove a regularity result: such spaces are $W^{1,p}_{\mathrm{loc}}\cap C^{0,α}_{\mathrm{loc}}$-Riemannian manifolds whose weight functions also belong to $W^{1,p}_{\mathrm{loc}}\cap C^{0,α}_{\mathrm{loc}}$, in both distance and harmonic charts for all $p<\infty$ and $α\in (0,1)$. We also establish a quantitative lower bound for the harmonic radius, together with compactness theorems and quantitative geometric bounds. As a byproduct, we develop a comprehensive elliptic regularity theory in this setting. These results apply, in particular, to smooth weighted Riemannian manifolds and to RCD spaces satisfying a synthetic curvature upper bound, or CBA condition. Even in these settings, the resulting regularity statements are new. In the latter case, both the Riemannian metric and the weight function are shown to be locally Lipschitz. As further applications, we establish fibration theorems and use metric smoothing to confirm, in a synthetic framework, a conjecture of V. Kapovitch concerning almost flat manifolds with mixed curvature bounds.
