과제정보
This work is partially supported by the NSF of China (Grant Nos. 11801496, 11926352 and 12261095), the Fok Ying-Tung Education Foundation (China) and Hubei Key Laboratory of Applied Mathematics (Hubei University).
참고문헌
- M. T. Anderson, L2 harmonic forms on complete Riemannian manifolds, in Geometry and analysis on manifolds (Katata/Kyoto, 1987), 1-19, Lecture Notes in Math., 1339, Springer, Berlin, 1988. https://doi.org/10.1007/BFb0083043
- G. P. Bessa and J. F. Montenegro, Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann. Global Anal. Geom. 24 (2003), no. 3, 279-290. https://doi.org/10.1023/A:1024750713006
- H.-D. Cao and H. Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (2013), no. 3-4, 879-889. https://doi.org/10.1007/s00526-012-0508-1
- H.-D. Cao, Y. Shen, and S.-H. 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. do Carmo, Q. Wang, and C. Y. Xia, Complete submanifolds with bounded mean curvature in a Hadamard manifold, J. Geom. Phys. 60 (2010), no. 1, 142-154. https://doi.org/10.1016/j.geomphys.2009.09.001
- G. Carron, L2 harmonic forms on non-compact Riemannian manifolds, in Surveys in analysis and operator theory (Canberra, 2001), 49-59, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002.
- I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984.
- S. Y. Cheng, Eigenfunctions and eigenvalues of Laplacian, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), 185-193, Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Amer. Math. Soc., Providence, RI, 1975.
- S. Y. Cheng, Liouville theorem for harmonic maps, in Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), 147-151, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980.
- Q.-M. Cheng and S. Ogata, 2-dimensional complete self-shrinkers in ℝ3, Math. Z. 284 (2016), no. 1-2, 537-542. https://doi.org/10.1007/s00209-016-1665-2
- Q.-M. Cheng and Y.-J. Peng, Complete self-shrinkers of the mean curvature flow, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 497-506. https://doi.org/10.1007/s00526-014-0720-2
- L.-F. Cheung and P. F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525-530. https://doi.org/10.1007/PL00004840
- T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755-833. https://doi.org/10.4007/annals.2012.175.2.7
- A. Dessai, Obstructions to positive curvature and symmetry, Adv. Math. 210 (2007), no. 2, 560-577. https://doi.org/10.1016/j.aim.2006.07.003
- Q. Ding and Y. L. Xin, The rigidity theorems of self-shrinkers, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5067-5085. https://doi.org/10.1090/S0002-9947-2014-05901-1
- J. Dodziuk, L2 -harmonic forms on complete manifolds, in Seminars on differential geometry, edited by S. T. Yau, Ann. Math. Stud. 102, 1982.
- F. Du and J. Mao, Estimates for the first eigenvalue of the drifting Laplace and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space, J. Math. Anal. Appl. 456 (2017), no. 2, 787-795. https://doi.org/10.1016/j.jmaa.2017.07.044
- J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), no. 5, 385-524. https://doi.org/10.1112/blms/20.5.385
- J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037
- F. Fang and S. X. Mendon,ca, Complex immersions in Kahler manifolds of positive holomorphic k-Ricci curvature, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3725-3738. https://doi.org/10.1090/S0002-9947-05-03675-5
- F. Fang, S. X. Mendon,ca, and X. Rong, A connectedness principle in the geometry of positive curvature, Comm. Anal. Geom. 13 (2005), no. 4, 671-695. https://doi.org/10.4310/CAG.2005.v13.n4.a2
- P. Freitas, J. Mao, and I. Salavessa, Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds, Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 701-724. https://doi.org/10.1007/s00526-013-0692-7
- K. R. Frensel, Stable complete surfaces with constant mean curvature, Bol. Soc. Brasil. Mat. (N.S.) 27 (1996), no. 2, 129-144. https://doi.org/10.1007/BF01259356
- M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178-215. https://doi.org/10.1007/s000390300004
- S. Hildebrandt, J. Jost, and K.-O. Widman, Harmonic mappings and minimal submanifolds, Invent. Math. 62 (1980/81), no. 2, 269-298. https://doi.org/10.1007/BF01389161
- 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
- Z. Jin, Liouville theorems for harmonic maps, Invent. Math. 108 (1992), no. 1, 1-10. https://doi.org/10.1007/BF02100594
- K. Kenmotsu and C. Xia, Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific J. Math. 176 (1996), no. 1, 129-139. http://projecteuclid.org/euclid.pjm/1102352054 102352054
- P. Li, Curvature and function theory on Riemannian manifolds, In: Survey in Differential Geometry, in Honor of Atiyah, Bott, Hirzebruch and Singer, Vol. VII, pp. 71-111, International Press, Cambridge, 2000.
- P. Li, Harmonic functions and applications to complete manifolds, XIV Escola de Geometria Diferencial., Instituto de Matem'atica Pura e Aplicada (IMPA), Rio de Janeiro, 2006.
- 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
- H. Li and Y. Wei, f-minimal surface and manifold with positive m-Bakry-Emery Ricci curvature, J. Geom. Anal. 25 (2015), no. 1, 421-435. https://doi.org/10.1007/s12220-013-9434-5
- W. Lu, J. Mao, C. Wu, and L. Zeng, Eigenvalue estimates for the drifting Laplacian and the p-Laplacian on submanifolds of warped products, Appl. Anal. 100 (2021), no. 11, 2275-2300. https://doi.org/10.1080/00036811.2019.1679793
- J. Mao, R. Tu, and K. Zeng, Eigenvalue estimates for submanifolds in Hadamard manifolds and product manifolds N × R, Hiroshima Math. J. 50 (2020), no. 1, 17-42. https://doi.org/10.32917/hmj/1583550013
- H. P. McKean Jr., An upper bound to the spectrum of Δ on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359-366. http://projecteuclid.org/euclid.jdg/1214429509 https://doi.org/10.4310/jdg/1214429509
- J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of ℝn, Comm. Pure Appl. Math. 26 (1973), 361-379. https://doi.org/10.1002/cpa.3160260305
- R. Miyaoka, L,2 harmonic 1-forms on a complete stable minimal hypersurface, in Geometry and global analysis (Sendai, 1993), 289-293, Tohoku Univ., Sendai, 1993.
- B. Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991), no. 2, 185-188. https://doi.org/10.1007/BF02566644
- S. Pigola, A. G. Setti, and M. Rigoli, Vanishing and finiteness results in geometric analysis, Progress in Mathematics, 266, Birkhauser Verlag, Basel, 2008. https://doi.org/10.1007/978-3-7643-8642-9
- R. Schoen and S.-T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), no. 3, 333-341. https://doi.org/10.1007/BF02568161
- R. Schoen and S.-T. Yau, Lectures on differential geometry, translated from the Chinese by Ding and S. Y. Cheng, with a preface translated from the Chinese by Kaising Tso, Conference Proceedings and Lecture Notes in Geometry and Topology, I, Int. Press, Cambridge, MA, 1994.
- Y. Shen, A Liouville theorem for harmonic maps, Amer. J. Math. 117 (1995), no. 3, 773-785. https://doi.org/10.2307/2375088
- Q. Wang and C. Xia, Complete submanifolds of manifolds of negative curvature, Ann. Global Anal. Geom. 39 (2011), no. 1, 83-97. https://doi.org/10.1007/s10455-010-9224-2
- G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. https://doi.org/10.4310/jdg/1261495336
- Y. Zhao, C. Wu, J. Mao, and F. Du, Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates, Rev. Mat. Complut. 33 (2020), no. 2, 389-414. https://doi.org/10.1007/s13163-019-00322-1