A Campanato Regularity Theory for Multi-Valued Functions with Applications to Minimal Surface Regularity Theory

Minter, Paul

doi:10.48550/arxiv.2108.03085

Cited by 1 publication

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“…From here we may apply standard Campanato-style arguments (see e.g. [Wic14, Lemma 4.3] or [Min21a]) to reach the desired conclusion for ρ = 1/2. In view of property (B5I), the claimed estimate for arbitrary ρ ∈ (0, 1) follows from the case ρ = 1/2.…”

Section: Appendix a Hausdorff Dimension Bound For The Branch Set Of C...mentioning

confidence: 99%

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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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Section: Appendix a Hausdorff Dimension Bound For The Branch Set Of C...mentioning

confidence: 99%

“…, Q, follows from this (taken with 2 − 2β in place of β) and an appropriate version of the Campanato lemma (e.g. as in [Wic14, Lemma 4.3] or [Min21a]). The conclusion for γ ∈ (1/2, 1) follows by applying the conclusion for γ = 1/2 to appropriately translated and rescaled v and using a covering argument.…”

Section: 1mentioning

confidence: 99%