The singular set of minimal surfaces near polyhedral cones

Colombo, Maria; Edelen, Nick; Spolaor, Luca

doi:10.48550/arxiv.1709.09957

Cited by 8 publications

(19 citation statements)

References 17 publications

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“…We also suspect that in light of the present work, extensions of [CES17] to more general polyhedral cones, allowing for certain 4-way and 5-way junctions, is possible in the codimension one stable setting.…”

Section: Discussionmentioning

confidence: 83%

“…Arguably more important than the final results of [Sim93] are the ideas used; the ideas developed by L. Simon, in particular the dichotomy in [Sim93, Lemma 2.1], have been further developed in recent years. Key examples of this include [CES17], [BK17], [Wic14a], [BW18], [BW19]. In all of these cases in order to prove the ǫ-regularity result it is necessary to deal with the possibility of density gaps, or if possible rule them out a priori.…”

mentioning

confidence: 99%

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The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density $\mathbf{\frac{5}{2}}$

Minter¹

2021

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Adapting the blow-up arguments in [Sim93] and [Wic14a], we prove a C 1,α regularity theorem for stable codimension one integral n-varifolds which are sufficiently close to a "classical cone" of density 5 2 , i.e., a stationary integral cone supported on at most 5 half-hyperplanes meeting along a common (n − 1)-dimensional axis with density 5 2 at the origin. Our result is the first of its kind for non-flat cones with multiplicity > 1 when branch points are present in the nearby varifold. The only additional hypothesis necessary for our result is geometric: ruling out singularities comprised of three C 1,α hypersurfaces-withboundary meeting (only) along their common boundary; the corresponding results when the density of the classical cone is 3 2 or 2 -in which case the cone has multiplicity one almost everywhere -have been previously established in [Sim93] and [Wic21] respectively. For such varifolds the present work completes the analysis of the singular set in the region where the density is < 3, up to a set which is countably (n − 2)-rectifiable. A key new ingredient in our work is establishing a C 1,α boundary regularity theory for two-valued C 1,α harmonic functions which arise as blow-ups of sequences of such varifolds; this is discussed more generally in the accompanying work [Min21]. Subject to appropriate conditions on lower density classical singularities and (expected) regularity theorems concerning lower density branch point singularities, our arguments have the potential to be iterated inductively to establish regularity results for stable codimension one integral varifolds near classical cones of density q + 1 2 at the origin.

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Section: Discussionmentioning

confidence: 83%

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confidence: 99%

The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density $\mathbf{\frac{5}{2}}$

Minter¹

2021

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“…Pick J so that γ + J = 1, and recall that 1 + β ∈ (γ + J , γ + J+1 ). By assumption, we have sptM ∩ B 1/2 = graph U ⊂a+q(C) (u), where u satisfies the estimates (6). Since we are concerned with M ∩ B 1/4 , ensuring δ 2 (C) is small, there is no loss in assuming (after scaling, translating, rotating) that sptM = graph C 1 (u).…”

Section: Corollaries Related Resultsmentioning

confidence: 99%

“…In certain particular cases one can use the topology of M to deduce singular structure on M i . For example, when M is a union of halfplanes, then [10] showed that M i ∩ B 1/2 are a C 1,α perturbation of the M. Similar results hold if M has tetrahedral singularities, and the M i have an associated "orientation structure" ( [6]); or when M is a union of two planes, and the M i are 2-valued graphs ( [3]).…”

Section: Introductionmentioning

confidence: 95%

Regularity of minimal surfaces near quadratic cones

Edelen¹,

Spolaor²

2019

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proved that every area-minimizing hypercone C having only an isolated singularity fits into a foliation of R n+1 by smooth, area-minimizing hypersurfaces asymptotic to C. In this paper we prove that if a stationary varifold M in the unit ball B 1 ⊂ R n+1 lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone C 3,3 ), then sptM ∩ B 1/2 is a C 1,α perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of M , but of any nearby minimal surface. Our result also implies the Bernsteintype result [12], which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.N.E. was supported in part by NSF grant DMS-1606492. L.S. was partially supported by NSF grant DMS-1810645.1 Strongly isolated here means that some tangent cone of M at the singularity is a multiplicity-one cone with an isolated singularity at 0.

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“…One such instance directly relevant to the present work is [Wic14, Sections 10-14] where a degenerate situation in considered, in which the base cone C 0 is a higher-multiplicity hyperplane; the analysis done in [Wic14] in this case leads to Theorem 2.1 below which plays an essential role in our proof of Theorem A. See also: [Min21b] which studies a situation where branch points of the nearby varifolds do exist away from the axis of C 0 ; [BK17] where the "no significant gaps" condition used in [Sim93] fails (see the proof of Theorem C above); [CES17] where the (multiplicity 1) base cone C 0 is singular away from its spine; and [KW13], [KW17], [KW21] where the "cone" C 0 is the graph (possibly with multiplicity > 1) of a multi-valued homogeneous harmonic function ϕ, with varying degrees of homogeneity, including degrees of homogeneity < 1.…”

Section: Theorem B (Sheeting Theorem (Theorem 33 [Wic14])mentioning

confidence: 99%

A Structure Theory for Stable Codimension 1 Integral Varifolds with Applications to Area Minimising Hypersurfaces mod p

Minter¹,

Wickramasekera²

2021

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For any given Q ∈ { 3 2 , 2, 5 2 , 3, . . .}, we establish a structure theory for the class SQ of stable codimension 1 integral varifolds admitting no classical singularities of density < Q. This theory comprises three main theorems which describe: (i) the nature of a varifold V ∈ SQ, for integer Q, when V is close to a flat disk of multiplicity Q; (ii) the nature of a varifold V ∈ SQ when V is close to a flat disk of integer multiplicity < Q; and (iii) the nature of a varifold V ∈ SQ when V is close, in a ball, to a stationary cone with vertex density Q and support the union of 3 or more half-hyperplanes meeting along a common axis. The main new result in the present work concerns (i) and gives in particular a description of V ∈ SQ near branch points of density Q, while results concerning (ii) and (iii), giving that V is embedded or has the structure of a classical singularity respectively, directly follow from parts of the previous work [Wic14] (and are reproduced in Part 2 below).These three theorems, taken with Q = p/2, are readily applicable to codimension 1 rectifiable area minimising currents mod p for any integer p ≥ 2, establishing local structural properties of such a current T as consequences of little information, namely the (easily checked) stability of the regular part of T and the fact that such a 1-dimensional singular (representative) current in R 2 consists of p rays meeting at a point. Specifically, it follows from (i) 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 over P of a p 2 -valued function with C 1,α regularity in a certain generalised sense; this settles a basic remaining open question in the study of the local structure of codimension 1 area minimising currents mod p near points with planar tangent cones, extending the cases p = 2 and p = 4 of the result (with classical C 1,α conclusions near y) which have been known since the 1970's from the De Giorgi-Allard regularity theory ([All72]) and the structure theory of White ([Whi79]) respectively. If P has multiplicity < p/2 (for p even or odd), it follows from (ii) that T is smoothly embedded near y, recovering a second wellknown theorem of White ([Whi84]). Finally, the main structure results obtained recently by De Lellis-Hirsch-Marchese-Spolaor-Stuvard ([DLHM + 21]) for such currents T all follow from (iii). The implication to mod p minimising currents of the structure theory for S p/2 is analogous to how the regularity theory for codimension 1 area minimising integral currents is a direct corollary of the regularity theory ([Wic14]) for S∞ = ∩QSQ (the class of stable codimension 1 integral varifolds with no classical singularities).

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The singular set of minimal surfaces near polyhedral cones

Cited by 8 publications

References 17 publications

The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density $\mathbf{\frac{5}{2}}$

The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density $\mathbf{\frac{5}{2}}$

Regularity of minimal surfaces near quadratic cones

A Structure Theory for Stable Codimension 1 Integral Varifolds with Applications to Area Minimising Hypersurfaces mod p

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