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arXiv 2602.17859math.DGmath.CO
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对Gromov填充面积猜想的离散视角

A discrete view of Gromov's filling area conjecture

Auburn University（奥本大学）

Joseph Briggs, Chris Wells

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10 pages, 4 figures
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AI总结：

本文提出对Gromov填充面积猜想的离散类比，通过图论工具获得离散界限并转换到连续设置，首次给出等距填充的定量下界。

摘要：

一个紧致度量表面M等距地填充一个闭合度量曲线C，当且仅当M的边界是C，并且对于所有x,y∈C=∂M，d_M(x,y)=d_C(x,y)；也就是说，M不引入边界上任意两点之间的'捷径'。Gromov在微分几何中于1983年提出的填充面积猜想指出，在所有能够等距填充Riemannian圆的紧致、可定向Riemannian表面上，具有最小表面积的是半球。Gromov证明了当M与盘面同胚时，这一结论成立。尽管Gromov的猜想在一些其他情况下已被验证，但完整的猜想仍未解决。在本文中，我们考虑Gromov问题的离散类比，这可能对研究图在任意曲面上的嵌入具有自然意义。利用标准图论工具，如Menger定理，我们获得了该离散类比的合理渐进行为界限。然后我们展示如何将这些离散界限转换到连续设置中，表明任何长度为2π的Riemannian圆的等距填充的表面积至少为1.36π（半球的表面积为2π）。这似乎是首次对Gromov问题的定量下界，适用于任意等距填充。

英文摘要：

A compact metric surface $M$ isometrically fills a closed metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for every $x,y\in C=\partial M$; that is, $M$ does not introduce any ``shortcuts'' between points on its boundary. Gromov's filling area conjecture in differential geometry from 1983 asserts that among all compact, orientable Riemannian surfaces which isometrically fill the Riemannian circle, the one with the smallest surface area is the hemisphere. Gromov demonstrated that this is indeed the case if $M$ is homeomorphic to the disk. While Gromov's conjecture has since been verified in some other cases, the full conjecture remains unresolved. In this paper, we consider a discrete analogue of Gromov's problem, which is likely natural to those who study graph embeddings on arbitrary surfaces. Using standard graph-theoretic tools, such as Menger's theorem, we obtain reasonable asymptotic bounds on this discrete variant. We then demonstrate how these discrete bounds can be translated to the continuous setting, showing that any isometric filling of the Riemannian circle of length $2π$ has surface area at least $1.36π$ (the hemisphere has surface area $2π$). This appears to be the first quantitative lower-bound on Gromov's problem that applies to arbitrary isometric fillings.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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