\(p\)-harmonic 1-forms on totally real submanifolds in complex space forms. (English) Zbl 07742039
This paper contains vanishing type theorems for \(L^Q p\)-harmonic 1-forms on totally real submanifolds in complex space forms. The motivation for this paper stems from similar results in literature on harmonic 1-forms on totally real submanifolds. Thus, the present paper extends existing results to \(p\)-harmonic 1-forms. Vanishing type results are obtained for minimal (and non-minimal) totally real submanifolds under some mild assumptions. Another main achievement of the paper is the statement and proof of a vanishing theorem for \(p\)-harmonic 1-forms on \(n\)-dimensional non-compact totally real submanifolds in a complex projective space, under the assumption that the \(L^n\)-norm of the tracefree second fundamental form is sufficiently small. The proofs of these results involve establishing several non-trivial estimates using Bochner’s formula, Sobolev’s, Hölder’s, Cauchy-Schwarz’s, and other standard inequalities in analysis.
Reviewer: Peter Olanipekun (Auckland)
MSC:
| 58A10 | Differential forms in global analysis |
| 58A14 | Hodge theory in global analysis |
| 32L20 | Vanishing theorems |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
Keywords:
complex space forms; complex projective spaces; p-harmonic forms; totally real submanifolds; vanishing theorems; minimal surfacesReferences:
| [1] | Carron, G., Une suite exacte en \(####\)-cohomologie, Duke Math J, 95, 2, 343-372 (1998) · Zbl 0951.58024 |
| [2] | Carron, G., \(####\)-cohomologie et inégalités de Sobolev, Math Annalen, 314, 4, 613-639 (1999) · Zbl 0933.35054 |
| [3] | Li, P.; Tam, LF., Harmonic functions and the structure of complete manifolds, J Differ Geom, 35, 2, 359-383 (1992) · Zbl 0768.53018 |
| [4] | Miyaoka, R., \(####\) harmonic 1-forms on a complete stable minimal hypersurface, Geom Global Anal, 289-293 (1993) · Zbl 0912.53042 |
| [5] | Cao, H.; Shen, Y.; Zhu, S., The structure of stable minimal hypersurfaces in \(####\), Math Res Lett, 4, 5, 637-644 (1997) · Zbl 0906.53004 |
| [6] | Li, P.; Wang, J., Minimal hypersurfaces with redfinite index, Math Res Lett, 9, 1, 95-103 (2002) · Zbl 1019.53025 |
| [7] | Li, P.; Wang, J., Stable minimal hypersurfaces in a nonnegatively curved manifold, J Reine Angew Math, 566, 215-230 (2004) · Zbl 1050.53049 |
| [8] | Ni, L., Gap theorems for minimal submanifolds in \(####\), Commun Anal Geom, 9, 3, 641-656 (2001) · Zbl 1020.53041 |
| [9] | Seo, K., Minimal submanifolds with small total scalar curvature in Euclidean space, Kodai Math J, 31, 1, 113-119 (2008) · Zbl 1147.53313 |
| [10] | Pansu, P., Cohomologie \(####\) et pincement, Comment Math Helv, 83, 2, 327-357 (2008) · Zbl 1142.53023 |
| [11] | Yeganefar, N., \(####\)-cohomology of negatively curved manifolds, Ark Mat, 43, 2, 427-434 (2005) · Zbl 1093.53039 |
| [12] | Li, XD., On the strong \(####\)-Hodge decomposition over complete Riemannian manifolds, J Funct Anal, 257, 11, 3617-3646 (2009) · Zbl 1196.58001 |
| [13] | Uhlenbeck, K., Regularity for a class of non-linear elliptic systems, Acta Math, 138, 219-240 (1977) · Zbl 0372.35030 |
| [14] | Hamburger, C., Regularity of differential forms minimizing degenerate elliptic functionals, J Reine Angew Math, 431, 7-64 (1992) · Zbl 0776.35006 |
| [15] | Beck, L.; Stroffolini, B., Regularity results for differential forms solving degenerate elliptic systems, Calc Var, 46, 3-4, 769-808 (2013) · Zbl 1266.35052 |
| [16] | Giaquinta, M.; Hong, MC., Partial regularity of minimizers of a functional involving forms and maps, Nonlinear Differ Equ Appl, 11, 4, 469-490 (2004) · Zbl 1120.35037 |
| [17] | Han, YB; Zhang, QY; Liang, MH., \(####\) p-harmonic 1-forms on locally conformally flat Riemannian manifolds, Kodai Math J, 40, 3, 518-536 (2017) · Zbl 1384.53038 |
| [18] | Dung, NT; Sung, CJ., Analysis of weighted p-harmonic forms and applications, Int J Math, 30, 10, Article id: 1950058 (2019) · Zbl 1429.53059 |
| [19] | Sil, S., Nonlinear Stein theorem for differential forms, Calc Var, 58, 4, Article id: 154 (2019) · Zbl 1416.35060 |
| [20] | Lin, HZ; Yang, BG., The p-eigenvalue estimates and \(####\) p-harmonic forms on submanifolds of Hadamard manifolds, J Math Anal Appl, 488, 1, Aritcle id: 124018 (2020) · Zbl 1446.58012 |
| [21] | Seto, S., The first nonzero eigenvalue of the p-Laplacian on differential forms, Pacific J Math, 309, 1, 213-222 (2020) · Zbl 07307886 |
| [22] | Chen, BY; Ogiue, K., On totally real submanifolds, Trans Amer Math Soc, 193, 257-266 (1974) · Zbl 0286.53019 |
| [23] | Choi, H.; Seo, K., \(####\) harmonic 1-forms on totally real submanifolds in a complex projective space, Ann Global Anal Geom, 57, 3, 383-400 (2020) · Zbl 1445.53046 |
| [24] | Cuong, DV, Dung, NT, Son, NTK. Rigidity results on totally real submanifolds in complex space forms. accepted by Kyoto J Math. · Zbl 07836812 |
| [25] | Hoffman, D.; Spruck, J., Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm Pure Appl Math, 27, 6, 715-727 (1974) · Zbl 0295.53025 |
| [26] | Michael, J.; Simon, LM., Sobolev and mean-value inequalities on generalized submanifolds of \(####\), Comm Pure Appl Math, 26, 3, 361-379 (1973) · Zbl 0256.53006 |
| [27] | Bessa, GP; Montenegro, JF., Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann Global Anal Geom, 24, 3, 279-290 (2003) · Zbl 1060.53063 |
| [28] | Leung, PF., An estimate on the Ricci curvature of a submanifold and some applications, Proc Am Math Soc, 114, 4, 1051-1061 (1992) · Zbl 0753.53003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
