과제정보
The author would like to express his thanks to Prof. Nguyen Thac Dung for drawing his attention to this topic and useful discussion during the preparation of this work. The author was partially supported by Vingroup Innovation Foundation VINIF under grant number VINIF.2019. ThS.18.
참고문헌
- A. C. Bezerra and Q. Wang, Rigidity theorems for minimal submanifolds in a hyperbolic space, Ann. Acad. Sci. Fenn. Math. 42 (2017), no. 2, 905-920. https://doi.org/10.5186/aasfm.2017.4253
- E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243-268. https://doi.org/10.1007/BF01404309
- H.-D. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfaces in ℝn+1, Math. Res. Lett. 4 (1997), no. 5, 637-644. https://doi.org/10.4310/MRL.1997.v4.n5.a2
- M. P. Cavalcante, H. Mirandola, and F. Vitorio, L2-harmonic 1-forms on submanifolds with finite total curvature, J. Geom. Anal. 24 (2014), no. 1, 205-222. https://doi.org/10.1007/s12220-012-9334-0
- H. Choi and K. Seo, Lp harmonic 1-forms on minimal hypersurfaces with finite index, J. Geom. Phys. 129 (2018), 125-132. https://doi.org/10.1016/j.geomphys.2018.03.006
- H. Choi and K. Seo, Lp harmonic 1-forms on totally real submanifolds in a complex projective space, Ann. Global Anal. Geom. 57 (2020), no. 3, 383-400. https://doi.org/10.1007/s10455-020-09705-w
- N. T. Dung and K. Seo, Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold, J. Math. Anal. Appl. 423 (2015), no. 2, 1594-1609. https://doi.org/10.1016/j.jmaa.2014.10.076
- D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in threemanifolds, Invent. Math. 82 (1985), no. 1, 121-132. https://doi.org/10.1007/BF01394782
- R. Gulliver, Index and total curvature of complete minimal surfaces, in Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), 207-211, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986. https://doi.org/10.1090/pspum/044/840274
- D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727. https://doi.org/10.1002/cpa.3160270601
- P. F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1061. https://doi.org/10.2307/2159628
- P. Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, 134, Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139105798
- P. Li and L.-F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359-383. http://projecteuclid.org/euclid.jdg/1214448079
- P. Li and J. Wang, Minimal hypersurfaces with finite index, Math. Res. Lett. 9 (2002), no. 1, 95-103. https://doi.org/10.4310/MRL.2002.v9.n1.a7
- P. Li and J. Wang, Stable minimal hypersurfaces in a nonnegatively curved manifold, J. Reine Angew. Math. 566 (2004), 215-230. https://doi.org/10.1515/crll.2004.005
- N. M. B. Neto, Q. Wang, and C. Xia, Rigidity of complete minimal hypersurfaces in a hyperbolic space, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 659-668. https://doi.org/10.5186/aasfm.2015.4036
- B. Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991), no. 2, 185-188. https://doi.org/10.1007/BF02566644
- S. Pigola, M. Rigoli, and A. G. Setti, Vanishing and finiteness results in geometric analysis, Progress in Mathematics, 266, Birkhauser Verlag, Basel, 2008.
- J. Simons, Minimal varieties in Riemannian manifolds, Ann. Math. 80 (1964), 1-21. https://doi.org/10.2307/1970488
- J. Tysk, Finiteness of index and total scalar curvature for minimal hypersurfaces, Proc. Amer. Math. Soc. 105 (1989), no. 2, 429-435. https://doi.org/10.2307/2046961
- Q. Wang, On minimal submanifolds in an Euclidean space, Math. Nachr. 261/262 (2003), 176-180. https://doi.org/10.1002/mana.200310120