# A general regularity theory for stable codimension 1 integral varifolds

### Abstract

We give a necessary and sufficient geometric structural condition, which we call the $\alpha$-Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities. The $\alpha$-Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded $C^{1, \alpha}$ hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral $n$-varifold on a smooth $(n+1)$-dimensional Riemannian manifold satisfies the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$, its singular set is empty if $n \leq 6$, discrete if $n =7$ and has Hausdorff dimension $\leq n-7$ if $n \geq 8$; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$. The $\alpha$-Structural Hypothesis on an $n$-varifold for any $\alpha \in (0, 1/2)$ is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing $(n-1)$-dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions.
An optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems.

• [AW] W. K. Allard, "On the first variation of a varifold," Ann. of Math., vol. 95, pp. 417-491, 1972.
@article {AW, MRKEY = {0307015},
AUTHOR = {Allard, William K.},
TITLE = {On the first variation of a varifold},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {95},
YEAR = {1972},
PAGES = {417--491},
ISSN = {0003-486X},
MRCLASS = {49F20},
MRNUMBER = {0307015},
MRREVIEWER = {M. Klingmann},
DOI = {10.2307/1970868},
ZBLNUMBER = {0252.49028},
}
• [A1] F. J. Almgren Jr., "Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem," Ann. of Math., vol. 84, pp. 277-292, 1966.
@article {A1, MRKEY = {0200816},
AUTHOR = {Almgren, Jr., F. J.},
TITLE = {Some interior regularity theorems for minimal surfaces and an extension of {B}ernstein's theorem},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {84},
YEAR = {1966},
PAGES = {277--292},
ISSN = {0003-486X},
MRCLASS = {53.04},
MRNUMBER = {0200816},
MRREVIEWER = {W. P. Ziemer},
DOI = {10.2307/1970520},
ZBLNUMBER = {0146.11905},
}
• [A] F. J. Almgren Jr., Almgren’s Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet’s Integral and the Regularity of Area Minimizing Rectifiable Currents up to Codimension Two, River Edge, NJ: World Scientific Publ. Co., Inc., 2000, vol. 1.
@book{A, MRKEY = {1777737},
AUTHOR = {Almgren, Jr., F. J.},
TITLE = {Almgren's Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area Minimizing Rectifiable Currents up to Codimension Two},
SERIES={World Scientific Monograph Series in Math.},
VOLUME={1},
PUBLISHER={World Scientific Publ. Co., Inc.},
YEAR={2000},
ZBLNUMBER = {0985.49001},
MRNUMBER = {1777737},
}
• [DG] E. De Giorgi, Frontiere Orientate di Misura Minima, Pisa: Editrice Tecnico Scientifica, 1961.
@book {DG, MRKEY = {0179651},
AUTHOR = {De Giorgi, Ennio},
TITLE = {Frontiere Orientate di Misura Minima},
SERIES = {Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61},
PUBLISHER = {Editrice Tecnico Scientifica},
YEAR = {1961},
PAGES = {57},
MRCLASS = {49.00 (53.04)},
MRNUMBER = {0179651},
MRREVIEWER = {W. H. Fleming},
ZBLNUMBER = {0296.49031},
}
• [F1] H. Federer, "The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension," Bull. Amer. Math. Soc., vol. 76, pp. 767-771, 1970.
@article {F1, MRKEY = {0260981},
AUTHOR = {Federer, Herbert},
TITLE = {The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical Society},
VOLUME = {76},
YEAR = {1970},
PAGES = {767--771},
ISSN = {0002-9904},
MRCLASS = {28.80 (26.00)},
MRNUMBER = {0260981},
MRREVIEWER = {J. E. Brothers},
DOI = {10.1090/S0002-9904-1970-12542-3},
ZBLNUMBER = {0194.35803},
}
• [F] H. Federer, Geometric Measure Theory, New York: Springer-Verlag, 1969, vol. 153.
@book {F, MRKEY = {0257325},
AUTHOR = {Federer, Herbert},
TITLE = {Geometric Measure Theory},
SERIES = {Grundlehren Math. Wissen.},
VOLUME={153},
PUBLISHER = {Springer-Verlag},
YEAR = {1969},
PAGES = {xiv+676},
MRCLASS = {28.80 (26.00)},
MRNUMBER = {0257325},
MRREVIEWER = {J. E. Brothers},
ZBLNUMBER = {0176.00801},
}
• [FW] W. H. Fleming, "On the oriented Plateau problem," Rend. Circ. Mat. Palermo, vol. 11, pp. 69-90, 1962.
@article {FW, MRKEY = {0157263},
AUTHOR = {Fleming, Wendell H.},
TITLE = {On the oriented {P}lateau problem},
JOURNAL = {Rend. Circ. Mat. Palermo},
FJOURNAL = {Rendiconti del Circolo Matematico di Palermo. Serie II},
VOLUME = {11},
YEAR = {1962},
PAGES = {69--90},
ISSN = {0009-725X},
MRCLASS = {53.04 (49.00)},
MRNUMBER = {0157263},
MRREVIEWER = {R. Osserman},
DOI = {10.1007/BF02849427},
ZBLNUMBER = {0107.31304},
}
• [HS] R. Hardt and L. Simon, "Boundary regularity and embedded solutions for the oriented Plateau problem," Ann. of Math., vol. 110, iss. 3, pp. 439-486, 1979.
@article {HS, MRKEY = {0554379},
AUTHOR = {Hardt, Robert and Simon, Leon},
TITLE = {Boundary regularity and embedded solutions for the oriented {P}lateau problem},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {110},
YEAR = {1979},
NUMBER = {3},
PAGES = {439--486},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {49F10 (49F20 53A10)},
MRNUMBER = {0554379},
MRREVIEWER = {Jo ao Lucas Marqu{ê}s Barbosa},
DOI = {10.2307/1971233},
ZBLNUMBER = {0457.49029},
}
• [I] T. Ilmanen, "A strong maximum principle for singular minimal hypersurfaces," Calc. Var. Partial Differential Equations, vol. 4, iss. 5, pp. 443-467, 1996.
@article {I, MRKEY = {1402732},
AUTHOR = {Ilmanen, T.},
TITLE = {A strong maximum principle for singular minimal hypersurfaces},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {4},
YEAR = {1996},
NUMBER = {5},
PAGES = {443--467},
ISSN = {0944-2669},
MRCLASS = {49Q05 (58E12)},
MRNUMBER = {1402732},
DOI = {10.1007/BF01246151},
ZBLNUMBER = {0863.49030},
}
• [M] C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, New York: Springer-Verlag, 1966, vol. 130.
@book {M, MRKEY = {0202511},
AUTHOR = {Morrey, Jr., Charles B.},
TITLE = {Multiple Integrals in the Calculus of Variations},
SERIES = {Grundlehren Math. Wissen.},
VOLUME={130},
PUBLISHER = {Springer-Verlag},
YEAR = {1966},
PAGES = {ix+506},
MRCLASS = {49.00 (00.00)},
MRNUMBER = {0202511},
MRREVIEWER = {M. Schechter},
ZBLNUMBER = {0142.38701},
DOI = {10.1007/978-3-540-69952-1},
}
• [Mo] M. P. Moschen, "Principio di massimo forte per le frontiere di misura minima," Ann. Univ. Ferrara Sez. VII, vol. 23, pp. 165-168 (1978), 1977.
@article {Mo, MRKEY = {0482508},
AUTHOR = {Moschen, Maria Pia},
TITLE = {Principio di massimo forte per le frontiere di misura minima},
JOURNAL = {Ann. Univ. Ferrara Sez. VII},
VOLUME = {23},
YEAR = {1977},
PAGES = {165--168 (1978)},
MRCLASS = {49F10 (35J99)},
MRNUMBER = {0482508},
MRREVIEWER = {Klaus Steffen},
ZBLNUMBER = {0384.49030},
}
• [RR] E. R. Reifenberg, "Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type," Acta Math., vol. 104, pp. 1-92, 1960.
@article {RR, MRKEY = {0114145},
AUTHOR = {Reifenberg, E. R.},
TITLE = {Solution of the {P}lateau {P}roblem for {$m$}-dimensional surfaces of varying topological type},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {104},
YEAR = {1960},
PAGES = {1--92},
ISSN = {0001-5962},
MRCLASS = {49.00},
MRNUMBER = {0114145},
MRREVIEWER = {W. H. Fleming},
DOI = {10.1007/BF02547186},
ZBLNUMBER = {0099.08503},
}
• [R] L. Rosales, "The geometric structure of solutions to the two-valued minimal surface equation," Calc. Var. Partial Differential Equations, vol. 39, iss. 1-2, pp. 59-84, 2010.
@article {R, MRKEY = {2659679},
AUTHOR = {Rosales, Leobardo},
TITLE = {The geometric structure of solutions to the two-valued minimal surface equation},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {39},
YEAR = {2010},
NUMBER = {1-2},
PAGES = {59--84},
ISSN = {0944-2669},
MRCLASS = {49Q05 (49Q15 53A10)},
MRNUMBER = {2659679},
MRREVIEWER = {Gian Paolo Leonardi},
DOI = {10.1007/s00526-009-0301-y},
ZBLNUMBER = {1195.49051},
}
• [SS] R. Schoen and L. Simon, "Regularity of stable minimal hypersurfaces," Comm. Pure Appl. Math., vol. 34, iss. 6, pp. 741-797, 1981.
@article {SS, MRKEY = {0634285},
AUTHOR = {Schoen, Richard and Simon, Leon},
TITLE = {Regularity of stable minimal hypersurfaces},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {34},
YEAR = {1981},
NUMBER = {6},
PAGES = {741--797},
ISSN = {0010-3640},
CODEN = {CPAMAT},
MRCLASS = {49F22 (53C42 58E15)},
MRNUMBER = {0634285},
MRREVIEWER = {F. J. Almgren, Jr.},
DOI = {10.1002/cpa.3160340603},
ZBLNUMBER = {0497.49034},
}
• [SJ] J. Simons, "Minimal varieties in riemannian manifolds," Ann. of Math., vol. 88, pp. 62-105, 1968.
@article {SJ, MRKEY = {0233295},
AUTHOR = {Simons, James},
TITLE = {Minimal varieties in riemannian manifolds},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {88},
YEAR = {1968},
PAGES = {62--105},
ISSN = {0003-486X},
MRCLASS = {53.04 (35.00)},
MRNUMBER = {0233295},
MRREVIEWER = {W. F. Pohl},
DOI = {10.2307/1970556},
ZBLNUMBER = {0181.49702},
}
• [S1] L. Simon, Lectures on Geometric Measure Theory, Canberra: Australian National University Centre for Mathematical Analysis, 1983, vol. 3.
@book {S1, MRKEY = {0756417},
AUTHOR = {Simon, Leon},
TITLE = {Lectures on Geometric Measure Theory},
SERIES = {Proc. Centre Math. Anal. Austral. Nat. Univ.},
VOLUME = {3},
PUBLISHER = {Australian National University Centre for Mathematical Analysis},
YEAR = {1983},
PAGES = {vii+272},
ISBN = {0-86784-429-9},
MRCLASS = {49-01 (28A75 49F20)},
MRNUMBER = {0756417},
MRREVIEWER = {J. S. Joel},
ZBLNUMBER = {0546.49019},
}
• [S4] L. Simon, "A strict maximum principle for area minimizing hypersurfaces," J. Differential Geom., vol. 26, iss. 2, pp. 327-335, 1987.
@article {S4, MRKEY = {0906394},
AUTHOR = {Simon, Leon},
TITLE = {A strict maximum principle for area minimizing hypersurfaces},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {26},
YEAR = {1987},
NUMBER = {2},
PAGES = {327--335},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {49F10 (35J60 49F20 53C42)},
MRNUMBER = {0906394},
MRREVIEWER = {Michael Gr{ü}ter},
URL = {http://projecteuclid.org/euclid.jdg/1214441373},
ZBLNUMBER = {0625.53052},
}
• [S] L. Simon, "Cylindrical tangent cones and the singular set of minimal submanifolds," J. Differential Geom., vol. 38, iss. 3, pp. 585-652, 1993.
@article {S, MRKEY = {1243788},
AUTHOR = {Simon, Leon},
TITLE = {Cylindrical tangent cones and the singular set of minimal submanifolds},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {38},
YEAR = {1993},
NUMBER = {3},
PAGES = {585--652},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {58E15 (49Q20)},
MRNUMBER = {1243788},
MRREVIEWER = {J. E. Brothers},
URL = {http://projecteuclid.org/euclid.jdg/1214454484},
ZBLNUMBER = {0819.53029},
}
• [S3] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Basel: Birkhäuser, 1996.
@book {S3, MRKEY = {1399562},
AUTHOR = {Simon, Leon},
TITLE = {Theorems on Regularity and Singularity of Energy Minimizing Maps},
SERIES = {Lectures Math. ETH Zürich},
PUBLISHER = {Birkhäuser},
YEAR = {1996},
PAGES = {viii+152},
ISBN = {3-7643-5397-X},
MRCLASS = {58E20 (35J60 49N60 58G03)},
MRNUMBER = {1399562},
MRREVIEWER = {Nathan Smale},
DOI = {10.1007/978-3-0348-9193-6},
ZBLNUMBER = {0864.58015},
}
• [SW1] L. Simon and N. Wickramasekera, "Stable branched minimal immersions with prescribed boundary," J. Differential Geom., vol. 75, iss. 1, pp. 143-173, 2007.
@article {SW1, MRKEY = {2282727},
AUTHOR = {Simon, Leon and Wickramasekera, Neshan},
TITLE = {Stable branched minimal immersions with prescribed boundary},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {75},
YEAR = {2007},
NUMBER = {1},
PAGES = {143--173},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C42 (49Q05 53A07)},
MRNUMBER = {2282727},
MRREVIEWER = {Giandomenico Orlandi},
URL = {http://projecteuclid.org/euclid.jdg/1175266256},
ZBLNUMBER = {1109.53064},
}
• [SoW] B. Solomon and B. White, "A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals," Indiana Univ. Math. J., vol. 38, iss. 3, pp. 683-691, 1989.
@article {SoW, MRKEY = {1017330},
AUTHOR = {Solomon, Bruce and White, Brian},
TITLE = {A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {38},
YEAR = {1989},
NUMBER = {3},
PAGES = {683--691},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {49F20},
MRNUMBER = {1017330},
MRREVIEWER = {Martin Fuchs},
DOI = {10.1512/iumj.1989.38.38032},
ZBLNUMBER = {0711.49059},
}
• [W1] N. Wickramasekera, "A rigidity theorem for stable minimal hypercones," J. Differential Geom., vol. 68, iss. 3, pp. 433-514, 2004.
@article {W1, MRKEY = {2144538},
AUTHOR = {Wickramasekera, Neshan},
TITLE = {A rigidity theorem for stable minimal hypercones},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {68},
YEAR = {2004},
NUMBER = {3},
PAGES = {433--514},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {53C24 (49Q05)},
MRNUMBER = {2144538},