December  2019, 39(12): 7031-7056. doi: 10.3934/dcds.2019243

The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

1. 

SISSA, Via Bonomea 265, Trieste, 34136, Italy

2. 

Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

3. 

Mathematisches Institut, Universität Leipzig, Augustus Platz 10, Leipzig, D04109, Germany

* Corresponding author: Guido De Philippis

Received  January 2019 Revised  February 2019 Published  June 2019

Fund Project: The work of G.D.P. is supported by the INDAM-grant "Geometric Variational Problems".

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

Citation: Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
References:
[1]

W. K. Allard, A characterization of the area integrand, in Symposia Mathematica, (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome, 1973), Academic Press, London, Volume XIV, 1974,429-444. Google Scholar

[2]

W. K. Allard, An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287-331.  doi: 10.1007/BF01394028.  Google Scholar

[3]

W. K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, in Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics, (eds. F. J. Allard ad W. K. Almgren Jr.), 44, 1986. doi: 10.1090/pspum/044/840267.  Google Scholar

[4] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[5]

G. De PhilippisA. De Rosa and F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148.  doi: 10.1002/cpa.21713.  Google Scholar

[6]

G. De Philippis and F. Maggi, Dimensional estimates for singular sets in geometric variational problems with free boundaries, J. Reine Angew. Math., 725 (2017), 217-234.  doi: 10.1515/crelle-2014-0100.  Google Scholar

[7]

L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Centre for Mathematical Analysis, Canberra, 3, 1983.  Google Scholar

[8]

L. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281-326.  doi: 10.4310/CAG.1993.v1.n2.a4.  Google Scholar

[9]

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., 38 (1989), 683-691.  doi: 10.1512/iumj.1989.38.38032.  Google Scholar

[10]

B. White, Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math., 88 (1987), 243-256.  doi: 10.1007/BF01388908.  Google Scholar

[11]

B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom., 33 (1991), 413-443.  doi: 10.4310/jdg/1214446325.  Google Scholar

[12]

B. White, Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom., 102 (2016), 501-535.  doi: 10.4310/jdg/1456754017.  Google Scholar

show all references

References:
[1]

W. K. Allard, A characterization of the area integrand, in Symposia Mathematica, (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome, 1973), Academic Press, London, Volume XIV, 1974,429-444. Google Scholar

[2]

W. K. Allard, An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287-331.  doi: 10.1007/BF01394028.  Google Scholar

[3]

W. K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, in Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics, (eds. F. J. Allard ad W. K. Almgren Jr.), 44, 1986. doi: 10.1090/pspum/044/840267.  Google Scholar

[4] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[5]

G. De PhilippisA. De Rosa and F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148.  doi: 10.1002/cpa.21713.  Google Scholar

[6]

G. De Philippis and F. Maggi, Dimensional estimates for singular sets in geometric variational problems with free boundaries, J. Reine Angew. Math., 725 (2017), 217-234.  doi: 10.1515/crelle-2014-0100.  Google Scholar

[7]

L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Centre for Mathematical Analysis, Canberra, 3, 1983.  Google Scholar

[8]

L. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281-326.  doi: 10.4310/CAG.1993.v1.n2.a4.  Google Scholar

[9]

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., 38 (1989), 683-691.  doi: 10.1512/iumj.1989.38.38032.  Google Scholar

[10]

B. White, Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math., 88 (1987), 243-256.  doi: 10.1007/BF01388908.  Google Scholar

[11]

B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom., 33 (1991), 413-443.  doi: 10.4310/jdg/1214446325.  Google Scholar

[12]

B. White, Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom., 102 (2016), 501-535.  doi: 10.4310/jdg/1456754017.  Google Scholar

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