A gradient estimate for the linearized translator equation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915879386611712 |
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| author | Choi, Kyeongsu Haslhofer, Robert Hershkovits, Or |
| author_facet | Choi, Kyeongsu Haslhofer, Robert Hershkovits, Or |
| contents | In this paper, we develop some analytic foundations for the linearized translator equation in $\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in $\mathbb{R}^4$ fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for $W_v$, namely for the derivative of the variation field $W$ in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling $Y_v$, namely the derivative of the profile function $Y$ in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for $W_τ$ as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling $Y_τ$ in the tip region. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07629 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A gradient estimate for the linearized translator equation Choi, Kyeongsu Haslhofer, Robert Hershkovits, Or Analysis of PDEs Differential Geometry In this paper, we develop some analytic foundations for the linearized translator equation in $\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in $\mathbb{R}^4$ fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for $W_v$, namely for the derivative of the variation field $W$ in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling $Y_v$, namely the derivative of the profile function $Y$ in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for $W_τ$ as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling $Y_τ$ in the tip region. |
| title | A gradient estimate for the linearized translator equation |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2509.07629 |