We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$, then knowledge of least areas circumscribed by simple closed curves $\gamma \subset \partial M$ uniquely determines the metric $g$, under some additional geometric assumptions. These are that $g$ is either a) $C^3$-close to Euclidean or b) sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves $\gamma\subset \partial M$. We also prove a corresponding local result: assuming only that $(M,g)$ has strictly mean convex boundary at a point $p\in\partial M$, we prove that knowledge of the least areas circumscribed by any simple closed curve $\gamma$ in a neighbourhood $U\subset \partial M$ of $p$ uniquely determines the metric near $p$. The proofs rely on finding the metric along a continuous sweep-out of $M$ by area-minimizing surfaces; they bring together ideas from the 2D-Calder\'on inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.

中文翻译：

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从最小区域确定黎曼度量

我们证明，如果 $(M,g)$ 是一个具有 $C^4$-光滑黎曼度量 $g$ 和平均凸边界 $\partial M$ 的拓扑 3 球，那么最小面积的知识由在一些额外的几何假设下，简单的闭合曲线 $\gamma\subset\partial M$ 唯一地确定了度量 $g$。这些是 $g$ 要么是 a) $C^3$-接近欧几里得，要么 b) 足够薄。事实上，我们需要的最小面积数据是用于更受限制的曲线类 $\gamma\subset\partial M$。我们还证明了相应的局部结果：仅假设 $(M,g)$ 在点 $p\in\partial M$ 处具有严格的平均凸边界，我们证明了任何简单闭合曲线 $ 所包围的最小面积的知识$p$ 的邻域 $U\subset \partial M$ 中的 \gamma$ 唯一地确定了 $p$ 附近的度量。证明依赖于通过面积最小化表面沿着 $M$ 的连续扫描找到度量；他们汇集了来自 2D-Calder 的关于逆问题、极小曲面理论和对伪微分方程系统的仔细分析的想法。