论文标题
外部paneitz操作员和$ q $ curvatures for Hypersurfaces
Extrinsic Paneitz operators and $Q$-curvatures for hypersurfaces
论文作者
论文摘要
对于任何Hypersurface $ m $的Riemannian歧管$ X $，最近的作品介绍了外部保形拉普拉斯人和外部$ Q $ curvatures的概念。在这里，我们为外部版本$ {\ bf p} _4 $的外部版本提供了明确的公式，并在一般维度中的相应的外部四阶$ q $ -c $ -curvature $ {\ bf q} _4 $。该结果涉及嵌入$ m^4 \ hookrightArrow x^5 $的一系列明显的局部保形不变术（根据Weyl张量和无跟踪的第二个基本形式定义）和非平凡的本地形式形式形式的形式不变的$ \ Mathcal $ \ Mathcal {C} $。反过来，我们将$ \ MATHCAL {C} $识别为两个本地共形不变的线性组合$ J_1 $和$ J_2 $。此外，可以用$ j_1 $和$ j_2 $的线性组合用嵌入$ m \ m \ hookrightarrow x $的明显本地保形不变式表示。最终，这将$ {\ bf Q} _4 $结构的非平凡部分减少为非平凡不变的$ J_1 $。对于关闭的$ m^4 \ hookrightArrow {\ mathbb r}^5 $，我们将$ j_i $的积分与Guven和Graham-Reichert的功能联系起来。此外，我们在一般背景中建立了hypersurface $ m^4 \ hookrightArrow x^5 $的Graham-Reichert功能的Deser-Schwimmer类型分解。在这种情况下，我们发现了另一种本地保形不变$ J_3 $。最后，我们为封闭的$ m^4 \ hookrightarrow x^5 $的单数Yamabe Energy提供了一个明确的公式。由此产生的显式公式表明，它与总外部四阶$ q $ curvature成正比。该观察结果证实了一个普遍事实的特殊情况，并作为我们主要结果的额外交叉检查。
For any hypersurface $M$ of a Riemannian manifold $X$, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvatures. Here we derive explicit formulas for the extrinsic version ${\bf P}_4$ of the Paneitz operator and the corresponding extrinsic fourth-order $Q$-curvature ${\bf Q}_4$ in general dimensions. This result involves a series of obvious local conformal invariants of the embedding $M^4 \hookrightarrow X^5$ (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant $\mathcal{C}$. In turn, we identify $\mathcal{C}$ as a linear combination of two local conformal invariants $J_1$ and $J_2$. Moreover, a linear combination of $J_1$ and $J_2$ can be expressed in terms of obvious local conformal invariants of the embedding $M \hookrightarrow X$. This finally reduces the non-trivial part of the structure of ${\bf Q}_4$ to the non-trivial invariant $J_1$. For closed $M^4 \hookrightarrow {\mathbb R}^5$, we relate the integrals of $J_i$ to functionals of Guven and Graham-Reichert. Moreover, we establish a Deser-Schwimmer type decomposition of the Graham-Reichert functional of a hypersurface $M^4 \hookrightarrow X^5$ in general backgrounds. In this context, we find one further local conformal invariant $J_3$. Finally, we derive an explicit formula for the singular Yamabe energy of a closed $M^4 \hookrightarrow X^5$. The resulting explicit formulas show that it is proportional to the total extrinsic fourth-order $Q$-curvature. This observation confirms a special case of a general fact and serves as an additional cross-check of our main result.
