November  2021, 41(11): 5291-5302. doi: 10.3934/dcds.2021077

A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing

Department of Mathematics, UC Irvine, 340 Rowland Hall, Irvine, CA 92697-3875, USA

* Corresponding author: Connor Mooney

Received  February 2021 Revised  March 2021 Published  November 2021 Early access  April 2021

Fund Project: The first author is supported by NSF grant DMS-1854788

We give a proof by foliation that the cones over $ \mathbb{S}^k \times \mathbb{S}^l $ minimize parametric elliptic functionals for each $ k, \, l \geq 1 $. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

Citation: Connor Mooney, Yang Yang. A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5291-5302. doi: 10.3934/dcds.2021077
References:
[1]

F. J. Almgren and Jr ., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math., 84 (1966), 277-292.  doi: 10.2307/1970520.

[2]

F. J. Almgren Jr.R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., 139 (1977), 217-265.  doi: 10.1007/BF02392238.

[3]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[4]

A. Davini, On calibrations for Lawson's cones, Rend. Sem. Mat. Univ. Padova, 11 (2004), 55-70. 

[5]

E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85. 

[6]

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

[7]

G. De Philippis and E. Paolini, A short proof of the minimality of the Simons cone, Rend. Sem. Mat. Univ. Padova, 121 (2009), 233-241.  doi: 10.4171/RSMUP/121-14.

[8]

M. DelgadinoF. MaggiC. Mihaila and R. Neumayer, Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals, Arch. Ration. Mech. Anal., 230 (2018), 1131-1177.  doi: 10.1007/s00205-018-1267-8.

[9]

K. Ecker and G. Huisken, A Bernstein result for minimal graphs of controlled growth, J. Differential Geom., 31 (1990), 397-400. 

[10]

W. Fleming, On the oriented Plateau problem, Rend. Circolo Mat. Palermo, 9 (1962), 69-89.  doi: 10.1007/BF02849427.

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.

[12]

H. Jenkins, On $2$-dimensional variational problems in parametric form, Arch. Ration. Mech. Anal., 8 (1961), 181-206.  doi: 10.1007/BF00277437.

[13]

H. B. Lawson Jr., The equivariant Plateau problem and interior regularity, Trans. Amer. Math. Soc., 173 (1972), 231-249.  doi: 10.1090/S0002-9947-1972-0308905-4.

[14]

C. Mooney, Entire solutions to equations of minimal surface type in six dimensions, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.2307/1969841.

[15]

F. Morgan, The cone over the Clifford torus in $\mathbb{R}^4$ is $\Phi$-minimizing, Math. Ann., 289 (1991), 341-354.  doi: 10.1007/BF01446576.

[16]

P. Simoes, A Class of Minimal Cones in $\mathbb{R}^n, n \geq 8$ that Minimize Area, Ph.D thesis, UC Berkeley, 1973.

[17]

L. Simon, Entire solutions of the minimal surface equation, J. Differential Geom., 30 (1989), 643-688. 

[18]

L. Simon, On some extensions of Bernstein's theorem, Math. Z., 154 (1977), 265-273.  doi: 10.1007/BF01214329.

[19]

J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.  doi: 10.2307/1970556.

[20]

B. White, The mathematics of F. J. Almgren, Jr., J. Geom. Anal., 8 (1998), 681-702.  doi: 10.1007/BF02922665.

show all references

References:
[1]

F. J. Almgren and Jr ., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math., 84 (1966), 277-292.  doi: 10.2307/1970520.

[2]

F. J. Almgren Jr.R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., 139 (1977), 217-265.  doi: 10.1007/BF02392238.

[3]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[4]

A. Davini, On calibrations for Lawson's cones, Rend. Sem. Mat. Univ. Padova, 11 (2004), 55-70. 

[5]

E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85. 

[6]

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

[7]

G. De Philippis and E. Paolini, A short proof of the minimality of the Simons cone, Rend. Sem. Mat. Univ. Padova, 121 (2009), 233-241.  doi: 10.4171/RSMUP/121-14.

[8]

M. DelgadinoF. MaggiC. Mihaila and R. Neumayer, Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals, Arch. Ration. Mech. Anal., 230 (2018), 1131-1177.  doi: 10.1007/s00205-018-1267-8.

[9]

K. Ecker and G. Huisken, A Bernstein result for minimal graphs of controlled growth, J. Differential Geom., 31 (1990), 397-400. 

[10]

W. Fleming, On the oriented Plateau problem, Rend. Circolo Mat. Palermo, 9 (1962), 69-89.  doi: 10.1007/BF02849427.

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.

[12]

H. Jenkins, On $2$-dimensional variational problems in parametric form, Arch. Ration. Mech. Anal., 8 (1961), 181-206.  doi: 10.1007/BF00277437.

[13]

H. B. Lawson Jr., The equivariant Plateau problem and interior regularity, Trans. Amer. Math. Soc., 173 (1972), 231-249.  doi: 10.1090/S0002-9947-1972-0308905-4.

[14]

C. Mooney, Entire solutions to equations of minimal surface type in six dimensions, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.2307/1969841.

[15]

F. Morgan, The cone over the Clifford torus in $\mathbb{R}^4$ is $\Phi$-minimizing, Math. Ann., 289 (1991), 341-354.  doi: 10.1007/BF01446576.

[16]

P. Simoes, A Class of Minimal Cones in $\mathbb{R}^n, n \geq 8$ that Minimize Area, Ph.D thesis, UC Berkeley, 1973.

[17]

L. Simon, Entire solutions of the minimal surface equation, J. Differential Geom., 30 (1989), 643-688. 

[18]

L. Simon, On some extensions of Bernstein's theorem, Math. Z., 154 (1977), 265-273.  doi: 10.1007/BF01214329.

[19]

J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.  doi: 10.2307/1970556.

[20]

B. White, The mathematics of F. J. Almgren, Jr., J. Geom. Anal., 8 (1998), 681-702.  doi: 10.1007/BF02922665.

Figure 1.  The dilations of $ \Sigma_{kl} $ foliate one side of $ C_{kl} $
Figure 2.  The solution curve is contained in the region $ R $ bounded by $ \Gamma_1, \, \Gamma_2 $ and $ \Gamma_3 $
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