{"title":"渐近平坦静态流形的刚性定理及其应用","authors":"Brian Harvie, Ye-Kai Wang","doi":"10.1090/tran/9134","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M Superscript n Baseline comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M^{n},g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with boundary and with dimension <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than 8\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n>8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M Superscript n Baseline comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M^{n},g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems.</p> <p>As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than 8\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n > 8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the natural condition of <italic>Schwarzschild stability</italic>. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22].</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":"4 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A rigidity theorem for asymptotically flat static manifolds and its applications\",\"authors\":\"Brian Harvie, Ye-Kai Wang\",\"doi\":\"10.1090/tran/9134\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper M Superscript n Baseline comma g right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(M^{n},g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with boundary and with dimension <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n greater-than 8\\\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">n>8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper M Superscript n Baseline comma g right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(M^{n},g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems.</p> <p>As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n greater-than 8\\\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">n > 8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the natural condition of <italic>Schwarzschild stability</italic>. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. 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引用次数: 0
摘要
本文研究了麦考密克建立的有边界且维数为 n > 8 n>8 的渐近平坦静态流形 ( M n , g ) (M^{n},g) 的 Minkowski 型不等式[Proc. Amer. Math. Soc. 146 (2018), pp.]首先,我们证明了任何达到相等且具有 CMC 或等势边界的渐近平坦静态 ( M n , g ) (M^{n},g) 与施瓦兹柴尔德流形的旋转对称区域是等距的。然后,我们应用保角技术为有界静态势的水平集推导出一种新的闵科夫斯基式不等式。总之,这些都为检测渐近平坦静态系统的旋转对称性提供了一种稳健的方法。作为应用,我们证明了在n > 8 n > 8维度中,在施瓦兹柴尔德稳定性的自然条件下,由施瓦兹柴尔德坐标球和欧几里得坐标球诱导的巴特尼克数据的静态度量扩展的全局唯一性。这概括了 Miao [Classical Quantum Gravity 22 (2005), pp.]我们还建立了具有小爱因斯坦-希尔伯特能量的等势光子面的唯一性。这与最近关于静态光子面和黑洞的其他唯一性结果进行了有趣的比较,例如,见 V. Agostiniani 和 L. Mazzieri [Comm. Math. Phys. 355 (2017),pp. 261-301],C. Cederbaum 和 G. J. Galloway [J. Math. Phys. 62 (2021),p. 22],以及 S. Raulot [Classical Quantum Gravity 38 (2021),p. 22]。
A rigidity theorem for asymptotically flat static manifolds and its applications
In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds (Mn,g)(M^{n},g) with boundary and with dimension n>8n>8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static (Mn,g)(M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems.
As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n>8n > 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22].
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