April  2021, 41(4): 1929-1937. doi: 10.3934/dcds.2020347

Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature

1. 

LAMFA-CNRS UMR 7352, Université de Picardie Jules Verne, Faculté des Sciences, 33, rue Saint-Leu, 80039 Amiens CEDEX 1, France

2. 

Universidad Autónoma de Madrid, Facultad de Ciencias, Ciudad Universitaria de Cantoblanco, Calle Francisco Tomás y Valiente, 7, 28049 Madrid, Spain

* Corresponding author: Jesús Ocáriz

Received  January 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution $ u $ for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution $ u $.

Citation: Alberto Farina, Jesús Ocáriz. Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1929-1937. doi: 10.3934/dcds.2020347
References:
[1]

L. AmbrosioE. Brué and D. Semola, Rigidity of the 1-Bakery-Émery inequality and sets of finite perimeter in RCD spaces, Geom. Funct. Anal., 29 (2019), 949-1001.  doi: 10.1007/s00039-019-00504-5.

[2]

M. T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom., 18 (1983), 701-721.  doi: 10.4310/jdg/1214438178.

[3]

L. CaffarelliN. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473.  doi: 10.1002/cpa.3160471103.

[4]

J. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. Geom., 2 (1984), 141-151.  doi: 10.1007/BF01876506.

[5]

A. FarinaL. Mari and E. Valdinoci, Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Comm. Partial Differential Equations, 38 (2013), 1818-1862.  doi: 10.1080/03605302.2013.795969.

[6]

A. FarinaB. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791. 

[7]

A. FarinaY. Sire and E. Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds with Euclidean coverings, Proc. Amer. Math. Soc., 140 (2012), 927-930.  doi: 10.1090/S0002-9939-2011-11241-3.

[8]

A. Farina and E. Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature, Adv. Math., 225 (2010), 2808-2827.  doi: 10.1016/j.aim.2010.05.008.

[9]

A. Farina and E. Valdinoci, A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature, Discrete Contin. Dyn. Syst., 30 (2011), 1139-1144.  doi: 10.3934/dcds.2011.30.1139.

[10]

L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684.  doi: 10.1002/cpa.3160380515.

[11]

P. Petersen, Riemannian Geometry, Springer, New York, 2006.

[12]

S. Pigola, M. Rigoli and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis, Birkhäuser Verlag, Basel, 2008.

[13]

A. Rato and M. Rigoli, Gradient bounds and Liouville's type theorems for the Poisson equation on complete Riemannian manifolds, Tohoku Math. J. (2), 47 (1995), 509-519.  doi: 10.2748/tmj/1178225458.

[14]

T. Sakai, Riemannian Geometry, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/mmono/149.

[15]

S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.  doi: 10.1002/cpa.3160280203.

show all references

References:
[1]

L. AmbrosioE. Brué and D. Semola, Rigidity of the 1-Bakery-Émery inequality and sets of finite perimeter in RCD spaces, Geom. Funct. Anal., 29 (2019), 949-1001.  doi: 10.1007/s00039-019-00504-5.

[2]

M. T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom., 18 (1983), 701-721.  doi: 10.4310/jdg/1214438178.

[3]

L. CaffarelliN. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math., 47 (1994), 1457-1473.  doi: 10.1002/cpa.3160471103.

[4]

J. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. Geom., 2 (1984), 141-151.  doi: 10.1007/BF01876506.

[5]

A. FarinaL. Mari and E. Valdinoci, Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Comm. Partial Differential Equations, 38 (2013), 1818-1862.  doi: 10.1080/03605302.2013.795969.

[6]

A. FarinaB. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791. 

[7]

A. FarinaY. Sire and E. Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds with Euclidean coverings, Proc. Amer. Math. Soc., 140 (2012), 927-930.  doi: 10.1090/S0002-9939-2011-11241-3.

[8]

A. Farina and E. Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature, Adv. Math., 225 (2010), 2808-2827.  doi: 10.1016/j.aim.2010.05.008.

[9]

A. Farina and E. Valdinoci, A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature, Discrete Contin. Dyn. Syst., 30 (2011), 1139-1144.  doi: 10.3934/dcds.2011.30.1139.

[10]

L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684.  doi: 10.1002/cpa.3160380515.

[11]

P. Petersen, Riemannian Geometry, Springer, New York, 2006.

[12]

S. Pigola, M. Rigoli and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis, Birkhäuser Verlag, Basel, 2008.

[13]

A. Rato and M. Rigoli, Gradient bounds and Liouville's type theorems for the Poisson equation on complete Riemannian manifolds, Tohoku Math. J. (2), 47 (1995), 509-519.  doi: 10.2748/tmj/1178225458.

[14]

T. Sakai, Riemannian Geometry, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/mmono/149.

[15]

S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.  doi: 10.1002/cpa.3160280203.

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