A structure theory for stable codimension 1 integral varifolds with applications to area minimising hypersurfaces mod 𝑝

Author:

Minter Paul,Wickramasekera Neshan

Abstract

For any Q { 3 2 , 2 , 5 2 , 3 , } Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \} , we establish a structure theory for the class S Q \mathcal {S}_Q of stable codimension 1 stationary integral varifolds admitting no classical singularities of density > Q >Q . This theory comprises three main theorems which describe the nature of a varifold V S Q V\in \mathcal {S}_Q when: (i) V V is close to a flat disk of multiplicity Q Q (for integer Q Q ); (ii) V V is close to a flat disk of integer multiplicity > Q >Q ; and (iii) V V is close to a stationary cone with vertex density Q 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 S Q V\in \mathcal {S}_Q near branch points of density Q 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 Q=p/2 , are readily applicable to codimension 1 rectifiable area minimising currents mod p p for any integer p 2 p\geq 2 , establishing local structure properties of such a current T T as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even p p , if T T has one tangent cone at an interior point y y equal to an (oriented) hyperplane P P of multiplicity p / 2 p/2 , then P P is the unique tangent cone at y y , and T T near y y is given by the graph of a p 2 \frac {p}{2} -valued function with C 1 , α 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 p=2 and p = 4 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 P has multiplicity > p / 2 > p/2 (for p p even or odd), it follows from case (ii) that T T is smoothly embedded near y 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 T all follow from case (iii).

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference30 articles.

1. On the radial behavior of minimal surfaces and the uniqueness of their tangent cones;Allard, William K.;Ann. of Math. (2),1981

2. World Scientific Monograph Series in Mathematics;Almgren, Frederick J., Jr.,2000

3. On the first variation of a varifold;Allard, William K.;Ann. of Math. (2),1972

4. Transverse singularities of minimal two-valued graphs in arbitrary codimension;Becker-Kahn, Spencer T.;J. Differential Geom.,2017

5. [BKW21] Spencer Becker-Kahn and Neshan Wickramasekera, A regularity theorem for stationary integral varifolds near multiplicity 2 planes, 2021, preprint.

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Excess decay for minimizing hypercurrents mod 2Q;Nonlinear Analysis;2024-10

2. The structure of stable codimension one integral varifolds near classical cones of density $$Q+1/2$$;Calculus of Variations and Partial Differential Equations;2023-11-20

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3