Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909870735753216 |
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| author | Bhattacharya, Arunima Shankar, Ravi Wall, Jeremy Yepez, Diego |
| author_facet | Bhattacharya, Arunima Shankar, Ravi Wall, Jeremy Yepez, Diego |
| contents | We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|Θ|\geq (n-2)\tfracπ{2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^α$ singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving $C^{2,α}$ estimates by exponentiating the arctangent operator into a concave one when $|Θ|\geq (n-2)\tfracπ{2}$ and $n>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22741 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase Bhattacharya, Arunima Shankar, Ravi Wall, Jeremy Yepez, Diego Analysis of PDEs Differential Geometry We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|Θ|\geq (n-2)\tfracπ{2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^α$ singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving $C^{2,α}$ estimates by exponentiating the arctangent operator into a concave one when $|Θ|\geq (n-2)\tfracπ{2}$ and $n>2$. |
| title | Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2510.22741 |