发表机构
University of Rostock（罗斯托克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在四维洛伦兹流形中建立面积约束Willmore曲面的定向临界理论，证明尖锐曲率不等式及刚性，并应用于Hawking拟局部能量的正性与单调性。
AI 中文摘要
我们在四维洛伦兹流形中发展了闭类空曲面的面积约束Willmore理论。由于Euler-Lagrange方程取值于法丛，我们引入沿给定法方向的定向临界性，并确定由法联络决定的一类联络相容方向。我们的主要结果是：在Einstein张量符号条件下，对于联络相容类空方向的临界性，成立尖锐不等式 $2\lambda|\Sigma|+\int_\Sigma |\vec H|^2\\,d\mu\leq 8\pi\chi(\Sigma)$。在主导能量条件和 $\lambda\geq0$ 下，这推出 $\int_\Sigma |\vec H|^2\\,d\mu\leq16\pi$。等号具有刚性：$\Sigma$ 是圆的且满足Minkowski因果菱形刚性。我们还证明了零临界方向的类似尖锐不等式。这些结果给出了Hawking拟局部能量的正性、刚性、平稳性和无穷小单调性陈述。超曲面表述将先前研究的Hawking曲面问题恢复为空间法方向的临界性。球对称、Kerr和FLRW时空中的显式例子区分了定向临界性与完全临界性，并显示了刚性理论中符号假设的必要性。
英文摘要
We develop an area-constrained Willmore theory for closed spacelike surfaces in four-dimensional Lorentzian manifolds. Since the Euler--Lagrange equation is normal-bundle-valued, we introduce directional criticality along a prescribed normal direction and identify a class of connection-compatible directions determined by the normal connection. Our main result is the sharp inequality $2λ|Σ|+\int_Σ|\vec H|^2,dμ\leq 8πχ(Σ)$ for criticality in a connection-compatible spacelike direction under an Einstein-tensor sign condition. Under the dominant energy condition and $λ\geq0$, this yields $\int_Σ|\vec H|^2,dμ\leq16π$. Equality is rigid: $Σ$ is round and satisfies Minkowski causal-diamond rigidity. We also prove an analogous sharp inequality for null critical directions. These results give positivity, rigidity, stationarity, and infinitesimal monotonicity statements for the Hawking quasi-local energy. A hypersurface formulation recovers the previously studied Hawking-surface problem as criticality in the spatial normal direction. Explicit examples in spherical symmetry, Kerr, and FLRW spacetimes distinguish directional from full criticality and show the necessity of the sign assumptions in the rigidity theory.
