Abstract
A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersson, L., Chruściel, P., Friedrich, H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations. Comm. Math. Phys. 149, 587–612 (1992)
Aviles, P.: A study of the singularities of solutions of a class of nonlinear elliptic partial differential equations. Comm. Partial Differ. Equ. 7, 609–643 (1982)
Aviles, P., McOwen, R.C.: Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J. Differ. Geom. 27, 225–239 (1988)
Aviles, P., McOwen, R.C.: Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds. Duke Math. J. 56, 395–398 (1988)
Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42, 271–297 (1989)
Caffarelli, L., Guan, P., Ma, X.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Comm. Pure Appl. Math. 60, 1769–1791 (2007)
Gao, L., Yau, S.-T.: The existence of negatively Ricci curved metrics on three-manifolds. Invent. Math. 85, 637–652 (1986)
Guan, B.: Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary, Int. Math. Res. Not. 2008, Article ID rnn 105, 25 pp. (2008)
Gursky, M., Streets, J., Warren, M.: Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary. Calc. Var. P. D. E. 41, 21–43 (2011)
Kennington, A.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34, 687–704 (1985)
Kichenassamy, S.: Boundary behavior in the Loewner–Nirenberg problem. J. Funct. Anal. 222, 98–113 (2005)
Korevaar, N., Mazzeo, R., Pacard, F., Schoen, R.: Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135, 233–272 (1999)
Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17, 37–91 (1987)
Li, Y., Zhu, M.: Yamabe type equations on three-dimensional Riemannian manifolds. Comm. Contemp. Math. 1, 1–50 (1999)
Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations, In: Contributions to Analysis, 245–272. Academic Press, New York (1974)
Lohkamp, J.: Metrics of negative Ricci curvature. Ann. Math. 140, 655–683 (1994)
Lohkamp, J.: The higher dimensional positive mass theorem II. arXiv: 1612.07505 (2016)
Mazzeo, R.: Regularity for the singular Yamabe problem. Indiana Univ. Math. J. 40, 1277–1299 (1991)
Mazzeo, R., Pacard, F.: Constant scalar curvature metrics with isolated singularities. Duke Math. J. 99, 353–418 (1999)
Mazzeo, R., Pollack, D., Uhlenbeck, K.: Moduli spaces of singular Yamabe metrics. J. Am. Math. Soc. 9, 303–344 (1996)
Schoen, R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Comm. Pure Appl. Math. 41, 317–392 (1988)
Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)
Schoen, R., Yau, S.-T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92, 47–71 (1988)
Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Cambridge, MA (1994)
Schoen, R., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities. arXiv: 1704.05490 (2017)
Witten, E.: A new proof of the positive energy theorem. Comm. Math. Phys. 80, 381–402 (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author acknowledges the support of NSF Grant DMS-1404596. The second author acknowledges the support of NSFC Grant 11571019.
Rights and permissions
About this article
Cite this article
Han, Q., Shen, W. On the Negativity of Ricci Curvatures of Complete Conformal Metrics. Peking Math J 4, 83–117 (2021). https://doi.org/10.1007/s42543-020-00028-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42543-020-00028-0