半线性超定问题、散度公式与几何不等式
Semilinear overdetermined problems, a divergence formula, and geometric inequalities
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中文总结 AI 辅助
本文研究满足Serrin型问题的黎曼流形边界面积界，推导相关散度公式，证明Minkowski型、面积-电荷不等式，推广Boucher-Gibbons-Horowitz经典结果。
中文摘要 AI 辅助
考虑满足Serrin型问题的n维紧致黎曼流形及其边界，我们证明了该边界面积的精确上下界。随后给出本文主要结果：给定黎曼流形上特殊向量场的散度公式，该公式与亚静态流形及相关度量（如V-静态流形、静态流形、静电流形）密切相关。我们证明了V-亚静态流形在不同Neumann边界条件下的Minkowski型不等式，还证明了紧致及非紧致静电流形的面积-电荷不等式。本文所给散度公式的主要应用推广了Boucher-Gibbons-Horowitz的经典结果：我们证明满足零收敛条件的静态时空渐近双曲空间必为双曲空间。
英文摘要
Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This divergence formula is closely related to sub-static manifolds and related metrics, e.g., $V$-static, static, and electrostatic manifolds. We show Minkowski-type inequalities for $V$-sub-static manifolds under different Neumann boundary conditions. Moreover, we prove an area-charge inequality for compact (and noncompact) electrostatic manifolds. The main application of the divergence formula presented in this work generalizes the classical result of Boucher--Gibbons--Horowitz: we prove that an asymptotically hyperbolic space of a static spacetime satisfying the null convergence condition must be the hyperbolic space.
发表机构
- Universidade de Brasília（巴西利亚大学）
- Universidade Federal do Oeste da Bahia（巴伊亚联邦大学）
- Instituto Federal de Educação, Ciência e Tecnologia（联邦教育、科学和技术学院）
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