局部有界积分曲率曲面的分析
Analysis on surfaces with locally bounded integral curvature
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中文总结 AI 辅助
该研究针对局部有界积分曲率（BIC）完备奇异曲面，证明其为无穷小希尔伯特型等性质，结合Dynkin型条件得到其双李普希茨等价曲面及全局分析结果。
中文摘要 AI 辅助
我们针对具有局部有界积分曲率（简称BIC曲面）的（可能非紧的）完备奇异曲面证明了若干分析结果。将这些曲面视为配备二维豪斯多夫测度的度量测度空间，我们证明它们是无穷小希尔伯特型、局部加倍的，且满足局部庞加莱不等式。特别地，这蕴含了Cheeger拉普拉斯算子存在联合赫尔德连续的热核。假设BIC曲面的曲率测度的负部满足Dynkin型条件，我们证明该曲面双李普希茨等价于曲率测度有下界的BIC曲面，进而得到前述结果的全局版本。
英文摘要
We prove several analytic results on (possibly noncompact) complete singular surfaces having locally bounded integral curvature (in short: BIC surfaces). Regarding these as metric measure spaces with the 2-dimensional Hausdorff measure, we show that these are infinitesimally Hilbertian, locally doubling and satisfy a local Poincaré inequality. In particular, this entails the existence of a jointly Hölder continuous heat kernel for the Cheeger Laplacian. Assuming that the negative part of the curvature measure of a BIC surface satisfies a Dynkin-type condition, we show that the surface is bi-Lipschitz equivalent to a BIC surface with a lower bounded curvature measure, entailing global variants of the aforementioned results.
