We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner–Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a new hypersurface conformal invariant which generalises to higher dimensions the important Willmore invariant of embedded surfaces. We call this the obstruction density. For even dimensional hypersurfaces it is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman–Graham obstruction tensor for Poincaré–Einstein metrics; in part this is achieved by exploiting Bernstein–Gel’fand–Gel’fand machinery. The solution to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham–Jennes–Mason–Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers determine an explicit holographic formula for the obstruction density.

中文翻译：

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通过边界 Loewner-Nirenberg-Yamabe 问题的共形超曲面几何

我们通过将嵌入超曲面视为共形紧凑流形的共形无穷大来开发一种新的方法来处理嵌入的超曲面的共形几何。这涉及到 Loewner-Nirenberg 类型的问题，即在内部找到一个既共形紧又具有恒定标量曲率的度量。我们的第一个结果是所有阶次的渐近解。这涉及日志术语。我们表明，其中第一个的系数是一个新的超曲面保角不变量，它将嵌入曲面的重要 Willmore 不变量推广到更高维度。我们称之为障碍密度。对于偶维超曲面，它是基本曲率不变量。我们使后一个概念精确，并表明在合适的意义上，障碍密度和无迹第二基本形式是唯一的此类不变量。我们还表明，这种对平滑度的阻碍是 Poincaré-Einstein 度量的 Fefferman-Graham 阻碍张量的标量密度模拟；这部分是通过利用 Bernstein-Gel'fand-Gel'fand 机制实现的。恒定标量曲率问题的解决方案提供了一个平滑的超曲面定义密度，该密度由嵌入到障碍物的阶数来规范确定。我们给出了两个关键应用：共形超曲面不变量的构造和共形微分算子的构造。特别地，我们提出了由保形嵌入规范确定的拉普拉斯算子的无限保形幂族。通常，这些非常依赖于嵌入，并且与偶维超曲面固有的 Graham-Jennes-Mason-Sparling 算子相反，所有阶都存在。