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<p>We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norms and which in addition is already hyperbolic in the thin part, then it admits a genuine Einstein metric of negative sectional curvature <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^{2,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-close to the almost Einstein metric. Importantly, the constant <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varepsilon _0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.</p>

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Effective Stability of Negatively Curved Einstein Metrics in Dimensions at Least Four

  • 期刊论文

  • 发表日期：

    2026年8月4日

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  • Frieder Jäckel

摘要

We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in \(C^0\) C 0 - and \(L^2\) L 2 -norms and which in addition is already hyperbolic in the thin part, then it admits a genuine Einstein metric of negative sectional curvature \(C^{2,\alpha }\) C 2 , α -close to the almost Einstein metric. Importantly, the constant \(\varepsilon _0\) ε 0 measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.