AI 中文总结
该研究用逆平均曲率流构造保归一化面积的Lipschitz压缩映射，在二维情形证明了E. Milman的压缩猜想，还重现了Lin、Wang和Xu的相关定理。
AI 中文摘要
我们采用逆平均曲率流构造保归一化面积的Lipschitz压缩映射，其思路与Kim和E. Milman的热流构造类似。该构造给出从圆球面到球内闭严格凸超曲面的压缩映射，以及从平圆盘到欧氏球内严格凸自由边界超曲面的压缩映射。在二维情形下，这证明了E. Milman关于黎曼二维球面的压缩猜想，并为边界测地曲率为1的非负曲率圆盘给出类似内蕴结果。这些映射还在实际Lipschitz阈值处给出双侧谱比较与映射层面的刚性，重现了Lin、Wang和Xu的定理。
英文摘要
We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions $n\geq3$ show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every $n\geq2$, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.
CommentsThis new version includes some new results from arXiv:2605.24705v2
