Minimal spheres and scalar curvature
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918515610484736 |
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| author | Talipov, Talant |
| author_facet | Talipov, Talant |
| contents | In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\ge Λ_0>0$. Then there exist four distinct embedded minimal two-spheres $Σ_1,\ldots,Σ_4\subset (S^3,g)$ such that $\operatorname{area}_{g}(Σ_i)\le 12π(i+1)/Λ_0$ for every $i=1,\ldots,4$. We apply this result to a problem posed by S.-T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered at the origin in $\mathbb R^4$. Haslhofer-Ketover proved that ellipsoids with one sufficiently large semi-axis contain at least one non-planar embedded minimal two-sphere. We prove that such ellipsoids contain at least three non-planar embedded minimal two-spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21607 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal spheres and scalar curvature Talipov, Talant Differential Geometry Metric Geometry In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\ge Λ_0>0$. Then there exist four distinct embedded minimal two-spheres $Σ_1,\ldots,Σ_4\subset (S^3,g)$ such that $\operatorname{area}_{g}(Σ_i)\le 12π(i+1)/Λ_0$ for every $i=1,\ldots,4$. We apply this result to a problem posed by S.-T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered at the origin in $\mathbb R^4$. Haslhofer-Ketover proved that ellipsoids with one sufficiently large semi-axis contain at least one non-planar embedded minimal two-sphere. We prove that such ellipsoids contain at least three non-planar embedded minimal two-spheres. |
| title | Minimal spheres and scalar curvature |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2605.21607 |