Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866914651622604800 |
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| author | White, Brian |
| author_facet | White, Brian |
| contents | For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces.
Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that $k<4π$. In this paper, we use mean curvature flow to show that $k >3π$. We get a slightly larger lower bound for orientable surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_06893 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries White, Brian Differential Geometry 53C44 For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that $k<4π$. In this paper, we use mean curvature flow to show that $k >3π$. We get a slightly larger lower bound for orientable surfaces. |
| title | Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries |
| topic | Differential Geometry 53C44 |
| url | https://arxiv.org/abs/2106.06893 |