Prescribing curvature problems on the Bakry-Emery Ricci tensor of a compact manifold with boundary

Abstract

The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Ampère type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.

Keywords

k-Curvature / Bakry-Emery Ricci tensor / Complete metric / Dirichlet problem

References

[1] Bakry D, Émery M. Diffusions Hypercontractives in Séminaire de Probabilités XIX, 1983/84, 1985, Berlin: Springer-Verlag 177-20

[2] Bakry D, Qian Z. Volume comparison theorems without Jacobi field in current trends in potential theory. Theta Ser. Adv. Math., 2005, 4: 115-122

[3] Caffarelli L A, Nirenberg L, Spruck J. The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of eigenvalues of the Hessians. Acta Mathematica, 1985, 155: 261-301

[4] Guan B. Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary. Int. Math. Res. Not., 2008 1-25

[5] Gursky M, Streets J, Warren M. Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary. Calc. Var. Partial Diff. Equ., 2011, 41(1–2): 21-43

[6] Gursky M, Viaclovsky J. Fully nonlinear equations on Riemannian manifolds with negative curvature. Indiana Univ. Math. J., 2003, 52: 399-419

[7] He Y, Sheng W M. Prescribing the symmetric function of the eigenvalues of the Schouten tensor. Proc. Amer. Math. Soc., 2011, 139(3): 1127-1136

[8] Ledoux M. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math., 2000, 9(6): 305-366

[9] Li J, Sheng W M. Deforming metrics with negative curvature by a fully nonlinear flow. Calc. Var. Partial Diff. Equ., 2005, 23: 33-50

[10] Li Q R, Sheng W M. Some Dirichlet problems arising from conformal geometry. Pacific J. Math., 2011, 251(2): 337-359

[11] Lohkamp J. Metrics of negative Ricci curvature. Ann. Math., 1994, 140: 655-683

[12] Lott J. Some geometric properties of the Barkry-Émery-Ricci tensor. Comment. Math. Helv., 2003, 78(4): 865-883

[13] Lott J, Villani C. Ricci curvature for metrici-measure spaces via optimal transport. Ann. Math., 2009, 169: 903-991

[14] Perelman G. The entropy formula for the Ricci flow and its geometric applications, 2002

[15] Sheng W M, Trudinger N S, Wang X J. The Yamabe problem for higher order curvatures. J. Diff. Geom., 2007, 77: 515-553

[16] Sturm K T. On the geometry of metric measure spaces, I, II. Acta Math., 2006, 196(1): 65-177

[17] Su Y H, Zhang H C. Rigidity of manifolds with Bakry-Emery Ricci curvature bounded below. Geom. Dedicata, 2012, 160: 321-331

[18] Trudinger N S, Wang X J. The intermediate case of the Yamabe problem for higher order curvatures. Int. Math. Res. Not., 2010, 2010(13): 2437-2458

[19] Urbas J. An expansion of convex hypersurfaces. J. Diff. Geom., 1991, 33: 91-125

[20] Wei G, Wylie W. Comparison geometry for the Bakry-Émery-Ricci tensor. J. Diff. Geom., 2009, 83(2): 337-405

[21] Yuan, L. X., Prescribing curvature problem of Bakry-Emery Ricci tensor, Sci. China Math., DOI: 10.1007/s11425-013-4595-z.