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<p>Let (<i>M</i>,<i>g</i><sub>0</sub>) be a closed oriented hyperbolic manifold of dimension at least 3. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric <i>g</i> on <i>M</i> with same volume as <i>g</i><sub>0</sub>, its volume entropy <i>h</i>(<i>g</i>) satisfies <i>h</i>(<i>g</i>)≥<i>n</i>−1 with equality only when <i>g</i> is isometric to <i>g</i><sub>0</sub>. We show that the hyperbolic metric <i>g</i><sub>0</sub> is stable in the following sense: if <i>g</i><sub><i>i</i></sub> is a sequence of Riemaniann metrics on <i>M</i> of same volume as <i>g</i><sub>0</sub> and if <i>h</i>(<i>g</i><sub><i>i</i></sub>) converges to <i>n</i>−1, then there are smooth subsets <i>Z</i><sub><i>i</i></sub>⊂<i>M</i> such that both <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_711_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\operatorname{Vol}(Z_{i},g_{i})$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_711_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\operatorname{Area}(\partial Z_{i},g_{i})$</EquationSource> </InlineEquation> tend to 0, and (<i>M</i>∖<i>Z</i><sub><i>i</i></sub>,<i>g</i><sub><i>i</i></sub>) converges to (<i>M</i>,<i>g</i><sub>0</sub>) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for <i>M</i> is intrinsically isomorphic to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_711_Article_IEq3.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$(M,\frac{(n-1)^{2}}{4n} g_{0})$</EquationSource> </InlineEquation>.</p>

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Entropy and Stability of Hyperbolic Manifolds

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  • 发表日期：

    2025年5月14日

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  • Antoine Song

摘要

Let (M,g0) be a closed oriented hyperbolic manifold of dimension at least 3. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric g on M with same volume as g0, its volume entropy h(g) satisfies h(g)≥n−1 with equality only when g is isometric to g0. We show that the hyperbolic metric g0 is stable in the following sense: if gi is a sequence of Riemaniann metrics on M of same volume as g0 and if h(gi) converges to n−1, then there are smooth subsets Zi⊂M such that both $\operatorname{Vol}(Z_{i},g_{i})$ and $\operatorname{Area}(\partial Z_{i},g_{i})$ tend to 0, and (M∖Zi,gi) converges to (M,g0) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for M is intrinsically isomorphic to $(M,\frac{(n-1)^{2}}{4n} g_{0})$ .