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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.19681 |
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| _version_ | 1866911462910328832 |
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| author | Impera, Debora Rimoldi, Michele Ruatta, Francesco |
| author_facet | Impera, Debora Rimoldi, Michele Ruatta, Francesco |
| contents | We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this assumption, such self-shrinkers reduce effectively to codimension one and are therefore generalized self-shrinking cylinders. In contrast to previous works, our approach is purely elliptic: it relies on parabolicity in a weighted setting and is tailored specifically to self-shrinkers, rather than to general ancient solutions of the flow. This allows us to avoid assuming any uniform pinching condition, to treat in any dimension the sharp Andrews-Baker pinching constant $\frac{4}{3n}$ and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of quadratically pinched self-shrinkers in higher codimension Impera, Debora Rimoldi, Michele Ruatta, Francesco Differential Geometry We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this assumption, such self-shrinkers reduce effectively to codimension one and are therefore generalized self-shrinking cylinders. In contrast to previous works, our approach is purely elliptic: it relies on parabolicity in a weighted setting and is tailored specifically to self-shrinkers, rather than to general ancient solutions of the flow. This allows us to avoid assuming any uniform pinching condition, to treat in any dimension the sharp Andrews-Baker pinching constant $\frac{4}{3n}$ and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions. |
| title | Classification of quadratically pinched self-shrinkers in higher codimension |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2602.19681 |