Abstract
We prove existence of integral rectifiable \(m\)-dimensional varifolds minimizing functionals of the type \(\int |H|^p\) and \(\int |A|^p\) in a given Riemannian \(n\)-dimensional manifold \((N,g),\,2\le m<n\) and \(p>m,\) under suitable assumptions on \(N\) (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in \({\mathbb{R }^S}\) involving \(\int |H|^p,\) to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allard, W.K.: On the first variation of a varifold. Ann. Math. 95, 417–491 (1972)
Almgren, F.J.: The Theory of Varifolds. In: Mimeographed Notes. Princeton, Princeton University Press (1965)
Anzellotti, G., Serapioni, R., Tamanini, I.: Curvatures, functionals, currents. Indiana Univ. Math. J. 39(3), 617–669 (1990)
Ambrosio, L., Gobbino, M., Pallara, D.: Approximation problems for curvature varifolds. J. Geom. Anal. 8(1), 1–19 (1998)
Brakke, K.: The Motion of a Surface by its Mean Curvature. In: Mathematical Notes. Princeton University Press, Princeton (1978)
Delladio, S.: Special generalized Gauss graphs and their application to minimization of functionals involving curvatures. J. Reine Angew. Math. 486, 17–43 (1997)
Duggan, J.P.: \(W ^{2, p}\) regularity for varifolds with mean curvature. Commun. Partial Diff. Equ. 11(9), 903–926 (1986)
Federer, H.: Geometric measure theory. Springer, New York (1969)
Hutchinson, J.E.: Second fundamental form for varifolds and the existence of surfaces minimizing curvature. Indiana Math. J. 35(1), 45–71 (1986)
Hutchinson, J.E.: \(C^{1,\alpha }\) Multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in \(L^p\). Proc. Symp. Pure Math. 44, 281–306 (1986)
Hutchinson, J.E.: Some regularity theory for curvature varifolds. In: Miniconference Geometry and Partial Differential Equations Proceedings of the Centre for Mathematical Analysis, vol. 12, pp 60–66. ANU, Canberra (1987)
Kuwert, E., Schätzle, R.: Removability of isolated singularities of Willmore surfaces. Ann. Math. 160(1), 315–357 (2004)
Lamm, T., Metzger, J., Schulze, F.: Foliations of asymptotically flat manifolds by surfaces of Willmore type. Math. Ann. 350, 1–78 (2011)
Leonardi, G.P., Masnou, S.: Locality of the mean curvature of rectifiable varifolds. Adv. Calc. Var. 2(1), 17–42 (2009)
Mantegazza, C.: Su Alcune Definizioni Deboli di Curvatura per Insiemi Non Orientati. Degree Thesis, University of Pisa, http://cvgmt.sns.it/cgi/get.cgi/papers/man93/thesis.pdf (1993) (Italian)
Mantegazza, C.: Curvature varifolds with boundary. J. Diff. Geom. 43, 807–843 (1996)
Menne, U.: Some applications of the isoperimetric inequality for integral varifolds. Adv. Calc. Var. 2, 247–269 (2009)
Menne, U.: Second order rectifiability of integral varifolds of locally bounded first variation. J. Geom. Anal. doi:10.1007/s12220-011-9261-5 (2011)
Meeks, W.H., Rosenberg, H.: Stable minimal surfaces in \(M\,\times \,{\mathbb{R}}\). J. Diff. Geom. 68(3), 515–534 (2004)
Meeks, W.H., Rosenberg, H.: The theory of minimal surfaces in \(M\,\times \,{\mathbb{R}}\). Commun. Math. Helv. 80(4), 811–858 (2005)
Mondino, A.: Some results about the existence of critical points for the Willmore functional. Math. Zeit. 266(3), 583–622 (2010)
Mondino, A.: The conformal Willmore functional: a perturbative approach. Preprint arXiv:1010.4151v1 (2010), J. Geom. Anal. (24 September 2011), pp. 1–48 (Online First)
Morgan, F.: Geometric Measure Theory, A Beginner’s Guide, 4th edn. Academic Press, Amsterdam (2009)
Moser, R.: A Generalization of Rellich’s Theorem and Regularity of Varifolds Minimizing Curvature. Max Planck Inst. Math. Natur. Leipzig, Preprint No 72 (2001)
Nelli, B., Rosenberg, H.: Minimal surfaces in \({\mathbb{H}}^2\,\times \,{\mathbb{R}}\). Bull. Braz. Math. Soc. 33(2), 263–292 (2002)
Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34(6), 741–797 (1981)
Simon, L.: Lectures on Geometric Measure Theory, vol. 3. Proceedings Center for Mathematics Analysis Australian National University, Canberra (1983)
Simon, L.: Existence of surfaces minimizing the Willmore functional. Commun. Anal. Geom. 1(2), 281–325 (1993)
Pitts, J.T.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. In: Mathematical Notes, vol. 27. Princeton University Press, Princeton (1981)
Rivière, T.: Analysis aspects of Willmore surfaces. Invent. Math. 174(1), 1–45 (2008)
White, B.: Which ambient spaces admit isoperimetric inequalities for submanifolds? J. Diff. Geom. 83(1), 213–228 (2009)
White, B.: The maximum principle for minimal varieties of arbitrary codimension. arXiv 0906.0189v2 (2010)
Willmore, T.J.: Riemannian Geometry. Oxford Science Publications, Oxford (1993)
Acknowledgments
This work has been supported by M.U.R.S.T under the Project FIRB-IDEAS “Analysis and Beyond”.The author would like to thank G. Bellettini, E. Kuwert, A. Malchiodi, C. Mantegazza, U. Menne and B. White for stimulating and fundamental discussions about the topics of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
Cite this article
Mondino, A. Existence of integral \(m\)-varifolds minimizing \(\int \!|A|^p\) and \(\int \!|H|^p,\,p>m,\) in Riemannian manifolds. Calc. Var. 49, 431–470 (2014). https://doi.org/10.1007/s00526-012-0588-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-012-0588-y