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Generic scarring for minimal hypersurfaces along stable hypersurfaces

Song, Antoine and Zhou, Xin (2021) Generic scarring for minimal hypersurfaces along stable hypersurfaces. Geometric and Functional Analysis, 31 (4). pp. 948-980. ISSN 1016-443X. doi:10.1007/s00039-021-00571-7.

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Let Mⁿ⁺¹ be a closed manifold of dimension 3 ≤ n + 1 ≤ 7. We show that for a C∞-generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface S ⊂ (M,g) corresponds a sequence of closed, embedded, minimal hypersurfaces {Σₖ} scarring along S, in the sense that the area and Morse index of Σₖ both diverge to infinity and, when properly renormalized, Σₖ converges to S as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian 3-manifods.

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Additional Information:This research was partially conducted during the period A.S. served as a Clay Research Fellow. X.Z. is partially supported by NSF Grants DMS-1811293, DMS-1945178, and an Alfred P. Sloan Research Fellowship. We would like to thank Peter Sarnak for discussions and for pointing out [BL67, Ral80].
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Clay Mathematics InstituteUNSPECIFIED
Alfred P. Sloan FoundationUNSPECIFIED
Issue or Number:4
Record Number:CaltechAUTHORS:20221026-539124000.5
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117593
Deposited By: George Porter
Deposited On:28 Oct 2022 15:35
Last Modified:28 Oct 2022 15:35

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