Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces

被引：12
作者：
De Rosa, Antonio [1 ]
Gioffre, Stefano [2 ]
机构：
[1] NYU, Courant Inst Math Sci, New York, NY 10012 USA
[2] Univ Zurich, Inst Math, Winterthurerstr 190, CH-8057 Zurich, Switzerland
来源：
JOURNAL OF GEOMETRIC ANALYSIS | 2019年 / 29卷 / 03期
关键词：
Umbilical hypersurfaces; Quantitative stability; Elliptic integrands; Wulff shape;
D O I：
10.1007/s12220-018-0079-2
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider n2, p(1, +) and sigma an n-dimensional, closed hypersurface in Rn+1, which is the boundary of a convex, open set. We show that if the Lp-norm of the trace-free part of the anisotropic second fundamental form is small, then sigma must be W2, p-close to the Wulff shape, with a quantitative estimate.
引用
收藏
页码：2318 / 2346
页数：29
相关论文
共 30 条
[1]  
[Anonymous], 1976, DIFFERENTIAL TOPOLOG
[2]  
[Anonymous], 1983, FUNDAMENTAL PRINCIPL
[3]   STABILITY OF HYPERSURFACES WITH CONSTANT MEAN-CURVATURE [J].
BARBOSA, JL ;
DOCARMO, M .
MATHEMATISCHE ZEITSCHRIFT, 1984, 185 (03) :339-353
[4]   A C0 estimate for nearly umbilical surfaces [J].
De Lellis, C ;
Müller, S .
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2006, 26 (03) :283-296
[5]  
De Lellis C, 2005, J DIFFER GEOM, V69, P75
[6]   Almost-Schur lemma [J].
De Lellis, Camillo ;
Topping, Peter M. .
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2012, 43 (3-4) :347-354
[7]   A direct approach to Plateau's problem in any codimension [J].
De Philippis, G. ;
De Rosa, A. ;
Ghiraldin, F. .
ADVANCES IN MATHEMATICS, 2016, 288 :59-80
[8]   Rectifiability of Varifolds with Locally Bounded First Variation with Respect to Anisotropic Surface Energies [J].
De Philippis, Guido ;
De Rosa, Antonio ;
Ghiraldin, Francesco .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2018, 71 (06) :1123-1148
[9]  
De Rosa A., 2018, ARXIV180309118
[10]   MINIMIZATION OF ANISOTROPIC ENERGIES IN CLASSES OF RECTIFIABLE VARIFOLDS [J].
De Rosa, Antonio .
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2018, 50 (01) :162-181
123