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Immersions with bounded second fundamental form. (English) Zbl 1318.53060

Summary: We first consider immersions on compact manifolds with uniform \(L^p\)-bounds on the second fundamental form and uniformly bounded volume. We show compactness in arbitrary dimension and codimension, generalizing a classical result of J. Langer [Math. Ann. 270, 223–234 (1985; Zbl 0564.58010)]. In the second part, this result is used to deduce a localized version, being more convenient for many applications, such as convergence proofs for geometric flows.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53B25 Local submanifolds

Citations:

Zbl 0564.58010
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References:

[1] Alt, H.W.: Lineare Funktionalanalysis, 4th edn. Springer, Berlin (2002) · Zbl 1098.46500
[2] Baker, C.: The mean curvature flow of submanifolds of high codimension. Ph.D. thesis (2010)
[3] Baker, C.: A partial classification of Type 1 singularities of the mean curvature flow in high codimension. Preprint (2011). arXiv:1104.4592 · Zbl 1254.53091
[4] Bauer, M., Kuwert, E.: Existence of Minimizing Willmore Surfaces of Prescribed Genus. Int. Math. Res. Not. 10, 553-576 (2003) · Zbl 1029.53073 · doi:10.1155/S1073792803208072
[5] Breuning, P.: Compactness of immersions with local Lipschitz representation. Ann. Inst. Henri Poincaré 29, 545-572 (2012) · Zbl 1254.53010 · doi:10.1016/j.anihpc.2012.02.001
[6] Breuning, P.: Immersions with local Lipschitz representation. Ph.D. thesis, Freiburg (2011) · Zbl 1219.53001
[7] Bröcker, T., Jänich, K.: Introduction to differential topology. Cambridge University Press, Cambridge (1982) · Zbl 0486.57001
[8] Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am. J. Math. 92, 61-74 (1970) · Zbl 0194.52902 · doi:10.2307/2373498
[9] Cooper, A.A.: A compactness theorem for the second fundamental form. Preprint (2011). arXiv:1006.5697v4 · Zbl 0694.53005
[10] Corlette, K.: Immersions with bounded curvature. Geom. Dedic. 33, 153-161 (1990) · Zbl 0717.53035 · doi:10.1007/BF00183081
[11] Delladio, S.: On Hypersurfaces in Rn+1 with Integral Bounds on Curvature. J. Geom. Anal. 11, 17-41 (2000) · Zbl 1034.49043 · doi:10.1007/BF02921952
[12] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992) · Zbl 0804.28001
[13] Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, second printing with corrections. Birkhäuser, Boston (2001)
[14] Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, New York (1976) · Zbl 0356.57001
[15] Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285-299 (1990) · Zbl 0694.53005
[16] Hutchinson, J.E.: Second Fundamental Form for Varifolds and the Existence of Surfaces Minimising Curvature. Indiana Univ. Math. J. 35(1), 45-71 (1986) · Zbl 0561.53008 · doi:10.1512/iumj.1986.35.35003
[17] Kuwert, E., Schätzle, R.: The Willmore flow with small initial energy. J. Differ. Geom. 57, 409-441 (2001) · Zbl 1035.53092
[18] Langer, J.: A Compactness Theorem for Surfaces with Lp-Bounded Second Fundamental Form. Math. Ann. 270, 223-234 (1985) · Zbl 0564.58010 · doi:10.1007/BF01456183
[19] Link, F.: Gradient Flow for the Willmore Functional in Riemannian Manifolds. Ph.D. thesis, Freiburg (2013). arXiv:1308.6055 · Zbl 1288.53002
[20] Mondino, A.: Existence of integral m-varifolds minimizing ∫|A|p and ∫|H|p, p>m, in Riemannian manifolds. Preprint (2010). arXiv:1010.4514v1 · Zbl 1282.49041
[21] Ndiaye, C.B., Schätzle, R.: A convergence theorem for immersions with L2-bounded second fundamental form. Rend. Semin. Mat. Univ. Padova 127, 235-247 (2012). doi:10.4171/RSMUP/127-12 · Zbl 1254.53091 · doi:10.4171/RSMUP/127-12
[22] Schlichting, A.: Mittlerer Krümmungsfluss fast konvexer Lipschitzflächen. Diploma thesis, Freiburg (2009) · Zbl 0564.58010
[23] Simon, L.: Lectures on Geometric measure theory. Proc. of the Centre for Math. Analysis, vol. 3. Australian National University, Canberra (1983) · Zbl 0546.49019
[24] Smith, G.: An Arzela-Ascoli theorem for immersed submanifolds. Ann. Fac. Sci. Toulouse 16(4), 817-866 (2007) · Zbl 1158.53046 · doi:10.5802/afst.1168
[25] Struwe, M.: Variational Methods, 4th edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (2008) · Zbl 1284.49004
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