Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-13T17:37:37.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Low Complexity Solutions of the Allen–Cahn Equation on Three-spheres

Part of: Existence theories

Published online by Cambridge University Press:  07 January 2019

Robert Haslhofer  and
Mohammad N. Ivaki
Robert Haslhofer
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Email: roberth@math.toronto.edum.ivaki@utoronto.ca
Mohammad N. Ivaki
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Email: roberth@math.toronto.edum.ivaki@utoronto.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this short note, we prove that on the three-sphere with any bumpy metric there exist at least two pairs of solutions of the Allen–Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.

Keywords

Allen–Cahn equationphase transitionsmall index

MSC classification

Primary: 49J35: Minimax problems
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 287 - 291
Copyright
© Canadian Mathematical Society 2018 

Footnotes

R. H. was partially supported by NSERC grant RGPIN-2016-04331 and a Connaught New Researcher Award. M. I. was supported by a Marsden Postdoctoral Fellowship. Both authors thank the Fields Institute for providing an excellent research environment during the thematic program on Geometric Analysis.

References

Allen, S. and Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening . Acta. Metall. 27(1979), 1084–1095.Google Scholar
Brendle, S. and Huisken, G., Mean curvature flow with surgery of mean convex surfaces in three-manifolds . Invent. Math. 203(2016), no. 2, 615–654. https://doi.org/10.1007/s00222-015-0599-3.Google Scholar
Buzano, R., Haslhofer, R., and Hershkovits, O., The moduli space of two-convex embedded spheres. 2016. arxiv:1607.05604.Google Scholar
Chodosh, O. and Mantoulidis, C., Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates. 2018. arxiv:1803.02716.Google Scholar
del Pino, M., Kowalczyk, M., and Wei, J., Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in ℝ3 . J. Differential Geom. 93(2013), no. 1, 67–131. https://doi.org/10.4310/jdg/1357141507.Google Scholar
Gaspar, P. and Guaraco, M., The Allen-Cahn equation on closed manifolds . Calc. Var. Partial Differential Equations 57(2018), no. 4, 57:101. https://doi.org/10.1007/s00526-018-1379-x.Google Scholar
Hiesmayr, F., Spectrum and index of two-sided Allen-Cahn minimal hypersurfaces. 2017. arxiv:1704.07738.Google Scholar
Haslhofer, R. and Kleiner, B., Mean curvature flow with surgery . Duke Math. J. 166(2017), no. 9, 1591–1626. https://doi.org/10.1215/00127094-0000008X.Google Scholar
Haslhofer, R. and Ketover, D., Minimal two-spheres in three-spheres. 2017. arxiv:1708.06567.Google Scholar
Ketover, D. and Liokumovich, Y., On the existence of unstable minimal Heegaard surfaces. 2017. arxiv:1709.09744.Google Scholar
Ketover, D., Marques, F., and Neves, A., The catenoid estimate and its geometric applications. 2016. arxiv:1601.04514.Google Scholar
Marques, F. and Neves, A., Rigidity of min-max minimal spheres in three-manifolds . Duke Math. J. 161(2012), no. 14, 2725–2752. https://doi.org/10.1215/00127094-1813410.Google Scholar
Pacard, F., The role of minimal surfaces in the study of the Allen-Cahn equation. Lecture notes from the Santalo Summer School Geometric Analysis, University of Granada, Spain, 2012.Google Scholar
Pacard, F. and Ritore, M., From constant mean curvature hypersurfaces to the gradient theory of phase transitions . J. Differential Geom. 64(2003), no. 3, 359–423. https://doi.org/10.4310/jdg/1090426999.Google Scholar
Savin, O., Phase transitions, minimal surfaces and a conjecture of De Giorgi . In: Current developments in mathematics 2009, Int. Press, Somerville, MA, 2010, pp. 59–113.Google Scholar
Smith, F., On the existence of embedded minimal 2-spheres in the 3-sphere endowed with an arbitrary Riemannian metric. Phd thesis, Supervisor: Leon Simon, University of Melbourne, 1982.Google Scholar
Tonegawa, Y., On stable critical points for a singular perturbation problem . Comm. Anal. Geom. 13(2005), no. 2, 439–459. https://doi.org/10.4310/CAG.2005.v13.n2.a7.Google Scholar
Tonegawa, Y., Applications of geometric measure theory to two-phase separation problems . Sugaku Expositions 21(2008), 178–196.Google Scholar
Tonegawa, Y. and Wickramasekera, N., Stable phase interfaces in the van der Waals-Cahn-Hilliard theory . J. Reine Angew. Math. 668(2012), 191–210.Google Scholar
White, B., The space of minimal submanifolds for varying Riemannian metrics . Indiana Univ. Math. J. 40(1991), no. 1, 161–200. https://doi.org/10.1512/iumj.1991.40.40008.Google Scholar
White, B., On the bumpy metrics theorem for minimal submanifolds . Amer. J. Math. 139(2017), no. 4, 1149–1155. https://doi.org/10.1353/ajm.2017.0029.Google Scholar