A structure theory for stable codimension 1 integral varifolds with applications to area minimising hypersurfaces mod $p$
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- by Paul Minter and Neshan Wickramasekera;
- J. Amer. Math. Soc. 37 (2024), 861-927
- DOI: https://doi.org/10.1090/jams/1032
- Published electronically: October 3, 2023
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Abstract:
For any $Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \}$, we establish a structure theory for the class $\mathcal {S}_Q$ of stable codimension 1 stationary integral varifolds admitting no classical singularities of density $<Q$. This theory comprises three main theorems which describe the nature of a varifold $V\in \mathcal {S}_Q$ when: (i) $V$ is close to a flat disk of multiplicity $Q$ (for integer $Q$); (ii) $V$ is close to a flat disk of integer multiplicity $<Q$; and (iii) $V$ is close to a stationary cone with vertex density $Q$ and supports the union of 3 or more half-hyperplanes meeting along a common axis. The main new result concerns (i) and gives in particular a description of $V\in \mathcal {S}_Q$ near branch points of density $Q$. Results concerning (ii) and (iii) directly follow from parts of previous work of the second author [Ann. of Math. (2) 179 (2014), pp. 843–1007].
These three theorems, taken with $Q=p/2$, are readily applicable to codimension 1 rectifiable area minimising currents mod $p$ for any integer $p\geq 2$, establishing local structure properties of such a current $T$ as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even $p$, if $T$ has one tangent cone at an interior point $y$ equal to an (oriented) hyperplane $P$ of multiplicity $p/2$, then $P$ is the unique tangent cone at $y$, and $T$ near $y$ is given by the graph of a $\frac {p}{2}$-valued function with $C^{1,\alpha }$ regularity in a certain generalised sense. This settles a basic remaining open question in the study of the local structure of such currents near points with planar tangent cones, extending the cases $p=2$ and $p=4$ of the result which have been known since the 1970’s from the De Giorgi–Allard regularity theory [Ann. of Math. (2) 95 (1972), pp. 417–491] [Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, 1961] and the structure theory of White [Invent. Math. 53 (1979), pp. 45–58] respectively. If $P$ has multiplicity $< p/2$ (for $p$ even or odd), it follows from case (ii) that $T$ is smoothly embedded near $y$, recovering a second well-known theorem of White [Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1986, pp. 413–427]. Finally, the main structure results obtained recently by De Lellis–Hirsch–Marchese–Spolaor–Stuvard [arXiv:2105.08135, 2021] for such currents $T$ all follow from case (iii).
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Bibliographic Information
- Paul Minter
- Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom
- Address at time of publication: Princeton University and the Institute for Advanced Study, Princeton, New Jersey, NJ 08544, United States
- ORCID: 0000-0001-8033-3339
- Email: pm6978@princeton.edu
- Neshan Wickramasekera
- Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom
- MR Author ID: 749149
- ORCID: 0009-0008-2669-2195
- Email: n.wickramasekera@dpmms.cam.ac.uk
- Received by editor(s): March 9, 2022
- Received by editor(s) in revised form: May 11, 2023
- Published electronically: October 3, 2023
- Additional Notes: The first author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.
- © Copyright 2023 American Mathematical Society
- Journal: J. Amer. Math. Soc. 37 (2024), 861-927
- MSC (2020): Primary 53A10
- DOI: https://doi.org/10.1090/jams/1032
- MathSciNet review: 4736529