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An inequality for length and volume in the complex projective plane

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Abstract

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer–Mrowka proof of the Thom conjecture.

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Acknowledgements

The author is grateful to Alex Nabutovsky for suggesting the question of possible inequalities relating 1-dimensional invariants and the volume of \({{\mathbb {C}}{\mathbb {P}}}^2\).

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Correspondence to Mikhail G. Katz.

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Katz, M.G. An inequality for length and volume in the complex projective plane. Geom Dedicata 213, 49–56 (2021). https://doi.org/10.1007/s10711-020-00567-x

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