arXiv 1303.31152013-03math.DG

Cartan-Hadamard猜想与《小王子》

The Cartan-Hadamard conjecture and The Little Prince

Benoît Kloeckner, Greg Kuperberg

备注：: v3: significant rewritting of some proofs, a mistake in the proof of the ball counter-example has been corrected

摘要：

推广的Cartan-Hadamard猜想说，如果$Ω$是具有截面曲率$K \le κ\le 0$的完备单连通黎曼$n$ -流形$M$中的具有固定体积的区域，则当$Ω$是具有常曲率$K=κ$的圆形$n$ -球时，$Ω$的边界具有最小可能的边界体积。$n=2$和$κ=0$是Weil的一个旧结果。本文给出了这个猜想在$κ=0$时在$n=2$和$n=4$维上的统一证明，以及$κ\textless{} 0$猜想的一个特例和$κ\textgreater{} 0$猜想的一个版本.我们的论点使用了一个新的解释，基于光传输，最佳运输，和线性规划，克罗克的$n=4$和$κ=0$的证明。推广到$n=4$和$κ\ne 0$是一个新的结果。正如Croke所做的，我们将曲率条件$K \le κ$放宽到一个较弱的candle条件$Candle(κ)$或$LCD(κ)$，我们还找到了Cartan-Hadamard猜想的一个简单版本的反例：对于每个$\varepsilon \textgreater{} 0$，存在具有$(1-\varepsilon)$ -收缩负曲率的黎曼3球$Ω$，边界体积由$\varepsilon$的一个函数所限定，并且具有任意大的体积。小王子的问题“它的证明成为更一般方法的一部分。

英文摘要：

The generalized Cartan-Hadamard conjecture says that if $Ω$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le κ\le 0$, then the boundary of $Ω$ has the least possible boundary volume when $Ω$ is a round $n$-ball with constant curvature $K=κ$. The case $n=2$ and $κ=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $κ=0$, and a special case of the conjecture for $κ\textless{} 0$ and a version for $κ\textgreater{} 0$. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for $n=4$ and $κ=0$. The generalization to $n=4$ and $κ\ne 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K \le κ$ to a weaker candle condition $Candle(κ)$ or $LCD(κ)$.We also find counterexamples to a naïve version of the Cartan-Hadamard conjecture: For every $\varepsilon \textgreater{} 0$, there is a Riemannian 3-ball $Ω$ with $(1-\varepsilon)$-pinched negative curvature, and with boundary volume bounded by a function of $\varepsilon$ and with arbitrarily large volume.We begin with a pointwise isoperimetric problem called "the problem of the Little Prince." Its proof becomes part of the more general method.

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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英文摘要：

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