在論文中，當Ricci曲率有下界，我們研究了完備黎曼流形上p-Laplacian的譜幾何和梯度估計的性質。首先，我們證明了拉普拉斯算子的分裂型定理。然後，我們給出與p-Laplacian的第一特徵值對應的正p-特徵函數的局部梯度估計。此外，我們證明了p-特徵函數的最優全局梯度估計。另一方面，我們首先推導出正的p-Laplacian熱方程解的Li-Yau型梯度估計。最後，我們證明了在一些不同的曲率假設下，p-Laplacian的第一個特徵值沿著Ricci-Bourguignon流幾乎是隨處嚴格單調遞增和可微分的。

In this thesis, we study the properties of the spectrum geometric and gradient estimates for p-Laplacian on complete Riemannian manifolds with the Ricci curvature is bounded from below. First, we prove a splitting type theorem for the Laplacian. Then, we give a local gradient estimate for the positive p-eigenfunction associated to the first eigenvalue of the p-Laplacian. Moreover, we show global sharp gradient estimates for p-eigenfunctions. On the other hand, we first derive a Li-Yau type gradient estimate for the positive solutions to the p-Laplacian heat equation. Finally, we prove that the first eigenvalue of the p-Laplacian is strictly monotone increasing and differentiable almost everywhere along with the RicciBourguignon flow under some different curvature assumptions.

Contents
Acknowledgement i
Abstract iii
1 Preliminaries 1
1.1 Bochner-Weitzenb¨ock formula and Comparison Theorems . . . . . . .1
1.2 Ends of a manifold and parabolicity . . . . . . . . . . . . . . . . . . . . . . . 6
2 Decay estimate and splitting type theorem for Laplacian 9
2.1 Decay estimate for Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Splitting type theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Local gradient estimates for p-Laplacian 40
3.1 Estimate the norm Lb1 for the gradient of p-eigenfunctions . . . . . .41
3.2 The Moser iteration technique and local gradient estimates for p-Laplacian . 59
4 Sharp gradient estimate and spectral rigidity for p-Laplacian 64
4.1 Sharp gradient estimate for p-Laplacian . . . . . . . . . . . . . . . . . . . . . 64
4.2 Structure at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Gradient estimates for the p-Laplacian heat equations 102
5.1 Gradient estimate for the p-Laplacian heat equation . . . . . . . . . . . 103
5.2 Sharp gradient estimate for the heat equation . . . . . . . . . . . . . . . . . 114
6 Monotonicity of eigenvalues of the p-Laplace operator under the RicciBourguignon flow 124
6.1 Monotonicity of eigenvalues of the p-Laplace operator . . . . . . . . .124
6.2 Monotonic quantities along Ricci-Bourguignon flow . . . . . . . . . . . 136
References 142

[1] T. Aubin, Métriques riemanniennes et courbure, J. Differential Geom. 4 (1970), 383-424.
[2] D. Bakry and Z. M. Qian, Volume comparison theorems without Jacobi fields, Current Trends in Potential Theory, Theta Ser. Adv. Math. 4, Theta, Bucharest, 2005, 115-122.
[3] G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 679-698.
[4] J.-P. Bourguignon, Ricci curvature and Einstein metrics, Global differential geometry and global analysis (D. Ferus, W. K¨uhnel, U. Simon and B. Wegner, eds.), Lecture Notes in Math. 838, Springer, Berlin, 1981, 42-63.
[5] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom. 69 (2005), no. 2, 217-278.
[6] K. Brighton, A Liouville-type Theorem for Smooth Metric Measure spaces, J. Geom. Anal. 23 (2013), no. 2, 562-570.
[7] S. Buckley and P. Koskela, Ends of metric measure spaces and Sobolev inequality, Math. Z. 252 (2006), no. 2, 275-285.
[8] P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. 15 (1982), no. 2, 213-230.
[9] E. Calabi, An extension of E.Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45-56.
[10] X. Cao, Eigenvalues of −∆ + R2 on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435-441.
[11] X. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4075-4078.
[12] X. Cao, S. Hou, J. Ling, Estimate and monotonicity of the first eigenvalue under the Ricci flow, Math. Ann. 354 (2012), no. 2, 451-463.
[13] H. Cao, Y. Shen, S. Zhu, The structure of stable minimal hypersurfaces in Rn+1, Math. Res. Lett. 4 (1997), no. 5, 637-644.
[14] G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The RicciBourguignon flow, Pacific J. Math. 287 (2017), no. 2, 337-370.
[15] S.Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z.
143 (1975), no. 3, 289-297.
[16] S. Y. Cheng, Liouville theorem for harmonic maps. Geometry of the Laplace operator, Proc. Sympos. Pure Math. Univ. Hawaii, Honolulu, Hawaii, 1979, 147-151.
[17] S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), 333-354.
[18] H. I. Choi, On the Liouville theorem for harmonic maps, Proc. Amer. Math. Soc. 85 (1982), no. 1, 91-94.
[19] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications, II: Analytic aspects, Mathematical Surveys and Monographs 144, American Mathematical Society, Providence, RI, 2008.
[20] H. T. Dung, N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, to appear Proc. Amer. Math. Soc.
[21] H. T. Dung, Monotonicity of eigenvalues of the p-Laplace operator under the RicciBourguignon flow, to appear in Kodai Math. J.
[22] N. T. Dung, Rigidity properties of smooth metric measure spaces via the weighted p-Laplacian, Proc. Amer. Math. Soc. 145 (2017), no. 3, 1287-1299.
[23] N. T. Dung, N. D. Dat, Weighted p-harmonic functions and rigidity of smooth metric measure spaces, J. Math. Anal. Appl. 443 (2016), no. 2, 959-980.
[24] N. T. Dung, N. N. Khanh, Q. A. Ngo, Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces, Manuscripta Math. 155 (2018), no. 3-4, 471-501.
[25] N. T. Dung, N. N. Khanh, T. C. Son, The number of cusps of complete Riemannian manifolds with finite volume, Taiwanese J. Math. 22 (2018), no. 6, 1403-1425.
[26] A. E. Fischer, An introduction to conformal Ricci flow, Class.Quantum Grav. 21 (2004), no. 3, 171-218.
[27] J. F. Grosjean, p-Laplace operator and diameter of manifolds, Ann. Glob. Anal. Geom. 28 (2005), no. 3, 257-270.
[28] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306.
[29] R. S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126.
[30] G. Huang, B. Ma, Gradient estimates and Liouville type theorems for a nonlinear elliptic equation, Arch. Math. 105 (2015), no. 5, 491-499.
[31] X. R. Jiang, Gradient estimate for a nonlinear heat equation on Riemannian manifolds, Proc. Amer. Math. Soc. 144 (2016), no. 8, 3635-3642.
[32] S. Kawai, N. Nakauchi, The first eigenvalue of the p-Laplacian on a compact Riemannian manifold, Nonlinear Anal. 55 (2003), no. 1-2, 33-46.
[33] B. Kotschwar and L. Ni, Gradient estimate for p-harmonic function, 1/H flow and an entropy formula, Ann. Sci. École. Norm. Sup. 42 (2009), no.1, 1-36.
[34] K. H. Lam, Spectrum of the Laplacian on manifolds with Spin (9) holonomy, Math.
Res. Lett. 15 (2008), no. 6, 1167- 1186.
[35] K. H. Lam, Results on a weighted Poincare inequality of complete manifolds, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5043-5062.
[36] X. D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. 84 (2005), no. 10, 1295-1361.
[37] P. Li, Harmonic functions and applications to complete manifolds, Preprint (available on the author’s homepage).
[38] P. Li, Geometric Analysis, Cambridge Studies in Advanced Mathematics 134, Cambridge University Press, 2012.
[39] P. Li, J. F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359-383.
[40] P. Li, J. Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), no. 3, 501-534
[41] P. Li, J. Wang, Complete manifolds with positive spectrum II, J. Differential Geom. 62 (2002), no. 1, 143-162.
[42] P. Li, J. Wang, Weighted Poincare inequality and rigidity of complete manifolds, Ann. Sci. Ecole Norm. Sup. ´ 39 (2006), no. 6, 921-982.
[43] P. Li, J. Wang, Connectedness at infinity of complete K¨ahler manifolds, Amer. J. Math. 131 (2009), no. 3, 771-817.
[44] P. Li, S. T. Yau, On the parabolic kernel of the Schr¨odinger operator, Acta Math. 156 (1986), no. 3-4, 153-201.
[45] Y. Li, Li-Yau-Hamilton estimates and Bakry-Émery Ricci curvature, Nonlinear Anal. 113 (2015), 1-32.
[46] F. H. Lin, Q. S. Zhang, On ancient solutions of the heat equations, to appear in Comm. Pure Appl. Math, arXiv:1712.0409.
[47] P. Lu, J. Qing, and Y. Zheng, A note on conformal Ricci flow, Pacific J. Math. 268 (2014), no. 2, 413-434.
48] L. Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Glob. Anal. Geom. 29 (2006), no. 3, 287-292.
[49] J. Mao, Monotonicity of the first eigenvalue of the Laplace and p-Laplace operators under forced mean curvature flow, J. Korean Math. Soc. 55 (2018), no. 6, 1435-1458.
[50] A. M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. 39 (2000), no. 8, 1051-1068.
[51] R. Moser, The inverse mean curvature flow and p-harmonic functions, J. Eur. Math. Soc. 9 (2007), no. 1, 77-83.
[52] A. Mukherjea and K. Pothoven, Real and functional analysis, Part A: Real analysis, 2nd ed., Mathematical Concepts and Methods in Science and Engineering 27, Plenum Press, New York, 1984.
[53] O. Munteanu, and J. Wang, Smooth metric measure spaces with nonnegative curvature, Comm. Anal. Geom. 19 (2011), no. 3, 451-486.
[54] O. Munteanu, and J. Wang, Analysis of the weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55-94.
[55] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series 289, Cambridge University Press, Cambridge, 2002.
[56] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices (1992), no. 2, 27-38.
[57] R. Schoen, S. T. Yau, Lectures on Differential Geometry, International Press, Boston, 1994.
[58] P. Souplet, Q. S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38 (2006), no. 6, 1045-1053.
[59] C.-J. A. Sung, J. Wang, Sharp gradient estimate and spectral rigidity for p-Laplacian, Math. Res. Lett. 21 (2014), no. 4, 885-904.
[60] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150.
[61] J. L. Vazquez, Smoothing and decay estimates for nonlinear diffusion equations: Equations of porous medium type, Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, 2006.
[62] L. F. Wang, A splitting theorem for the weighted measure, Ann. Glob. Anal. Geom. 42 (2012), no. 1, 79-89.
[63] L. F. Wang, On Lp f-spectrum and τ-quasi-Einstein metric, J. Math. Anal. Appl. 389 (2012), no. 1, 195-204.
[64] X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8 (2001), no. 5-6, 671-688.
[65] X. Wang, On the L2-cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2002), no. 2, 311-327.
[66] X. Wang, L. Zhang, Local gradient estimate for p-harmonic functions on Riemannian manifolds, Comm. Anal. Geom. 19 (2011), no. 4, 759-771.
[67] Y. Wang, J. Yang and, W. Chen, Gradient estimates and entropy formulae for weighted p-heat equations on smooth metric measure spaces, Acta Math. Sci 33B (2013), no. 4, 963-974.
[68] G. F. Wei, W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405.
[69] J. Y. Wu, Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds, J. Math. Anal. Appl. 369 (2010), no. 1, 400-407.
[70] J. Y. Wu, First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 8, 1591-1598.
[71] J. Y. Wu, E.-M. Wang and Y. Zheng, First eigenvalue of the p-Laplace operator along the Ricci flow, Ann. Global Anal. Geom. 38 (2010), no. 1, 27-55.
[72] J. Y. Wu, Splitting theorems on complete manifolds with Bakry-Emery curvature, arXiv: math.DG/1112.6302.
[73] J. Y. Wu, A note on the splitting theorem for the weighted measure, Ann. Glob. Anal. Geom. 43 (2013), no. 3, 287-298.
[74] J. Y. Wu, Lp-Liouville theorems on complete smooth metric measure spaces, Bull. Sci. Math. 138 (2014), no. 4, 510-539.
[75] J. Y. Wu, Elliptic gradient estimates for a weighted heat equation and applications, Math. Z, 280 (2015), no. 1-2, 451-468.
[76] J. Y. Wu Elliptic gradient estimates for a nonlinear heat equation and applications, Nonlinear Anal. 151 (2017), 1-17.
[77] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228.
[78] R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35-50.
[79] L. Zhao, The first eigenvalue of the p-Laplace operator under powers of the mth mean curvature flow, Results Math. 63 (2013), no. 3-4, 937-948.
[80] F. Zeng, Q. He and B. Chen, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Pacific J. Math. 296 (2018), no. 1, 1-20.