Abstract
Let (M, g) be a compact Riemannian manifold of dimension n ≥3, and let Γ be a nonempty closed subset of M. The negative case of the Singular Yamabe Problem concerns the existence and behavior of a complete metric g on M∖Γ that has constant negative scalar curvature and is pointwise conformally related to the smooth metric g. Previous results have shown that when Γ is a smooth submanifold (with or without boundary) of dimension d, there exists such a metric if and only if \(d > \frac{{n - 2}}{2}\). In this paper, we consider a more general class of closed sets and show the existence of a complete conformai metric ĝ with constant negative scalar curvature which depends on the dimension of the tangent cone to Γ at every point. Specifically, provided Γ admits a nice tangent cone at p, we show that when the dimension of the tangent cone to Γ at p is less than \(\frac{{n - 2}}{2}\) then there cannot exist a negative Singular Yamabe metric ĝ on M∖Γ.
Similar content being viewed by others
References
Aubin, T.Nonlinear Analysis on Manifolds, Springer-Verlag, Berlin, 1982.
Aviles, P. A study of the singularities of solutions of a class of nonlinear elliptic partial differential equations,Comm. PDE,7, 609–643, (1982).
Aviles, P. and McOwen, R. Conformai deformation to constant negative scalar curvature in complete Riemannian manifolds,J. Biff. Geom.,27, 225–238, (1988).
Aviles, P. and McOwen, R. Complete conformai metrics with negative scalar curvature in compact Riemannian manifolds,Duke Math. J.,56, 395–397, (1988).
Baras, P. and Pierre, M. Singularités éliminables pour des equations semi-linéaires,Ann. Inst. Fourier Grenoble,34, 185–206, (1984).
Bianchi, M. and Rigoli, M. Positive solutions to Yamabe-type equations on complete manifolds, to appear, inTrans. AMS.
Calabi, E. An extension of E. Hopf's maximum principle with an application to Riemannian geometry,Duke Math. J.,25, 45–56, (1958).
Cheng, S.Y. and Yau, S.T. Differential equations on Riemannian manifolds and their geometric applications,Comm. Pure. Applied. Math.,XXVIII, 333–355, (1975).
Delanoë, P. Generalized stereographic projections with prescribed scalar curvature, inContemporary Mathematics: Geometry, Physics, and Nonlinear PDE. Oliker, V. and Treibergs, A., Eds., AMS, 1990.
Federer, H.Geometric Measure Theory, Springer-Verlag, Berlin, 1969.
Finn, D. Positive solutions of Δgu =u q +Su singular at submanifolds with boundary,Indiana Univ. Math. J.,43, 1359–1397, (1994).
Finn, D. Positive solutions to nonlinear elliptic equation with prescribed singularities, Ph.D. Thesis, Northeastern University, 1995.
Finn, D. Noncompact manifolds of constant negative scalar curvature and positive singular solutions of semilinear elliptic equations, preprint.
Finn, D. and McOwen, R. Singularities and asymptotics for the equation Δgu−u q =Su, Indiana Univ. Math. J.,42, 1487–1523, (1993).
Gilbarg, D. and Trudinger, N.Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
Goresky, M. and MacPherson, R.Stratified Morse Theory, Springer-Verlag, Berlin, 1988.
Kazdan,J. Prescribing the Curvature of a Riemannian Manifold, AMS, 1985.
Lee, J. and Parker, T. The Yamabe problem,Bull. Am. Math. Soc.,17, 37–91, (1987).
Li, P., Tam, L., and Yang, D. On the elliptic equation Δu+ ku −Ku p = 0 on complete Riemannian manifolds and their geometric applications I, to appear inTrans. AMS.
Loewner, C. and Nirenberg, L. Partial differential equations invariant under conformai or projective transformations, inContributions to Analysis, Academic Press, 245–275, 1975.
Mazzeo, R. Regularity for the singular Yamabe problem,Indiana Univ. Math. J.,40, 1227–1299, (1991).
Mazzeo, R. and Pacard, F. A construction of singular solutions for a semilinear equation using asymptotic analysis,J. Diff. Geom.,44, 331–370, (1996).
Mazzeo, R. and Pacard, F. Constant scalar curvature metrics with isolated singularities, preprint.
Mazzeo, R., Pollack, D., and Uhlenbeck, K. Moduli spaces of singular Yamabe metrics,J. Am. Math. Soc.,9(2), 303–344, (1996).
McOwen, R. Singularities and the conformai scalar curvature equation, inGeometric Analysis and Nonlinear PDE, Bakelma, I., Ed., Marcel Dekker, 1992.
McOwen, R.Partial Differential Equations: Methods and Applications, Prentice Hall, Englewood Cliffs, NJ, 1996.
Milnor, J.Singular Points of Complex Hypersurfaces, Princeton University Press, Princeton, NJ, 1968.
Morgan, F.Geometric Measure Theory: A Beginner's Guide, Academic Press, 1988.
Pacard, F. The Yamabe problem on subdomains of even dimensional spheres,Topol. Methods Nonlinear Anal.,6(1), 137–150, (1995).
Royden, H.L.Real Analysis, Macmillan, 1988.
Schoen, R. Conformai deformation of a Riemannian metric to constant scalar curvature,J. Diff. Geom,20, 479–495, (1984).
Schoen, R. The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation,Comm. Pure. Applied. Math.,XLI, 317–392, (1988).
Schoen, R. and Yau, S.T. On the structure of manifolds with positive curvature,Manu. Math.,28, 159–183, (1979).
Schoen, R. and Yau, S.T. Conformally flat manifolds, Kleinian groups and scalar curvature,Invent. Math.,92, 47–71, (1988).
Stone, D. The exponential map at an isolated singular point,Mem. Am. Math. Soc.,35(256), 184, (1982).
Thorn, R. Ensembles et morphisms stratifiés,Bull. Am. Math. Soc.,75, 240–284, (1969).
Veron, L. Singularités éliminables d'équations elliptiques non linéaires,J. Diff. Eq.,41, 87–95, (1981).
Whitney, H. Local properties of analytic varieties, inDifferential and Combinatorial Topology, Cairns, S., Ed., Princeton University Press, Princeton, NJ, 205–244, 1965.
Whitney, H. Tangents to an analytic variety,Ann. Math.,81, 496–549, (1965).
Author information
Authors and Affiliations
Additional information
Communicated by Steven Krantz
Rights and permissions
About this article
Cite this article
Finn, D.L. On the negative case of the Singular Yamabe Problem. J Geom Anal 9, 73–92 (1999). https://doi.org/10.1007/BF02923089
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02923089