<p>In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen–Cahn min–max procedure in Bellettini and Wickramasekera (<a href="http://arxiv.org/abs/2010.05847">arXiv:2010.05847</a>, 2020) (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1956_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1956_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>; the minimal case (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1956_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .</p>

On Isolated Singularities and Generic Regularity of Min–Max CMC Hypersurfaces

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2025年3月18日

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Costante Bellettini,
Kobe Marshall-Stevens

摘要

In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen–Cahn min–max procedure in Bellettini and Wickramasekera (arXiv:2010.05847, 2020) (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real $\lambda $ $λ$ we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature $\lambda $ $λ$ ; the minimal case ( $\lambda = 0$ $λ = 0$ ) of this result was obtained in Chodosh et al. (Ars Inveniendi Analytica, 2022) .