Skip to main content
SpringerLink
Log in

\(\sigma _{2}\) Yamabe problem on conic 4-spheres

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We raise the \(\sigma _2\) Yamabe problem for conic 4-manifolds. For conic 4-spheres, we find a necessary condition for the existence of solutions, and discuss the borderline case. This is a non-linear generalization of conic surface theory of Troyanov.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)

    MathSciNet  MATH  Google Scholar 

  2. Bartolucci, D.: On the best pinching constant of conformal metrics on \(S^{2}\) with one and two conical singularities. J. Geom. Anal. 23, 855–877 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bartolucci, D., De Marchis, F., Malchiodi, A.: Supercritical conformal metrics on surfaces with conical singularities. Int. Math. Res. Not. 24, 5625–5643 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bartolucci, D., De Marchis, F.: On the Ambjorn–Olesen electroweak condensates. J. Math. Phys. 53(7), 073704 (2012)

    Article  MathSciNet  Google Scholar 

  5. Branson, T., Gover, R.: Variational status of a class of fully nonlinear curvature prescription problems. Calc. Var. PDEs 32, 253–262 (2008)

    Article  MathSciNet  Google Scholar 

  6. Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \(-u=V(x)e^{u}\) in two dimensions. Comm. Partial Differ. Equ. 16, 1223–1253 (1991)

    Article  Google Scholar 

  7. Brendle, S., Viaclovsky, J.: A variational characterization for \(\sigma _{n/2}\). Calc. Var. PDEs 20, 399–402 (2004)

    Article  Google Scholar 

  8. Brothers, J., Ziemer, W.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Chen, X., Donaldson, S., Song, S.: Kähler–Einstein metrics on Fano manifolds I: Approximation of metrics with cone singularities. J. Am. Math. Soc. 28, 183–197 (2015)

    Article  Google Scholar 

  10. Chen, X., Donaldson, S., Song, S.: Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2. J. Am. Math. Soc 28, 199–234 (2015)

    Article  Google Scholar 

  11. Chen, X., Donaldson, S., Song, S.: Kähler–Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2 and completion of the main proof. J. Am. Math. Soc 28, 235–278 (2015)

    Article  Google Scholar 

  12. Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)

    Article  MathSciNet  Google Scholar 

  13. Chen, C.C., Lin, C.S.: A sharp sup+inf inequality for a nonlinear elliptic equation in \({\mathbb{R}}^{2}\). Commun. Anal. Geom. 6, 1–19 (1998)

    Article  Google Scholar 

  14. Chen, W., Li, C.: Prescribing Gaussian curvatures on surfaces with conical singularities. J. Geom. Anal. 1, 359–372 (1991)

    Article  MathSciNet  Google Scholar 

  15. Chen, W., Li, C.: What kinds of singular surfaces can admit constant curvature? Duke Math. J. 78(2), 437–451 (1995)

    Article  MathSciNet  Google Scholar 

  16. Chang, S.-Y.A., Han, Z.C., Yang, P.: Classification of singular radial solutions to the \(\sigma _{k}\) Yamabe equation on annular domains. JDE 216, 482–501 (2005)

    Article  MathSciNet  Google Scholar 

  17. Chang, S.-Y.A., Han, Z.C., Yang, P.: On the prescribing \(\sigma _{2}\) curvature equation on \(S^{4}\). Calc. Var. Partial Differ. Equ. 40, 539–565 (2011)

    Article  Google Scholar 

  18. Chang, S.-Y.A., Hang, F., Yang, P.: On a class of locally conformally flat manifolds. Int. Math. Res. Not. 4, 185–209 (2004)

    Article  MathSciNet  Google Scholar 

  19. Chang, S.-Y.A., Gursky, M., Yang, P.: An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. Math. 155, 711–789 (2002)

    Article  MathSciNet  Google Scholar 

  20. Chang, S.-Y.A., Gursky, M., Yang, P.: A prior estimate for a class of nonlinear equations on 4-manifolds. J. D’Analyse J. Math. 87, 151–186 (2002). (special issue in memory of Thomas Wolff)

    Article  MathSciNet  Google Scholar 

  21. Chang, S.-Y.A., Gursky, M., Yang, P.: Entire Solutions of a Fully Nonlinear Equation. Lectures on Partial Differential Equations, pp. 43–60. International Press, Vienna (2002)

    Google Scholar 

  22. Chang, S.-Y.A., Gursky, M., Yang, P.: A conformally invariant sphere theorem in four dimension. Publ. Math. Inst. Ht. Etud. Sci. 98, 105–143 (2003)

    Article  MathSciNet  Google Scholar 

  23. Chang, S.-Y.A., Qing, J., Yang, P.: On a conformal gap and finiteness theorem for a class of four-manifolds. Geom. Funct. Anal. 17, 404–434 (2007)

    Article  MathSciNet  Google Scholar 

  24. Chen, S.: Local estimates for some fully nonlinear elliptic equations. Int. Math. Res. Not. 55, 3403–3425 (2005)

    Article  MathSciNet  Google Scholar 

  25. Fang, H., Lai, M.: On convergence to a football. Math. Ann. 366, 83–100 (2016)

    Article  MathSciNet  Google Scholar 

  26. Fang, H., Lai, M.: On curvature pinching of conic 2-spheres. Calc. Var. Partial Differ. Equ. 55, 118 (2016)

    Article  MathSciNet  Google Scholar 

  27. Fang, H., Lai, M.: Volume bounds of conic 2-spheres. Int. J. Math. 29(2), 1850010 (2018)

    Article  MathSciNet  Google Scholar 

  28. Fang, H., Ma, B.: Constant Q curvature metric on conic 4-manifolds (preprint)

  29. Guan, B.: Conformal metrics with prescribed curvature function on manifolds with boundary. Am. J. Math. 129, 915–942 (2007)

    Article  MathSciNet  Google Scholar 

  30. Guan, P., Wang, G.: Local estimates for a class of fully nonlinear equations arising from conformal geometry. Int. Math. Res. Not. 2003(26), 1413–1432 (2003)

    Article  MathSciNet  Google Scholar 

  31. Guan, P., Wang, G.: A fully nonlinear conformal flow on locally conformally flat manifolds. J. fur die reine und angewandte Mathematik 557, 219–238 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Guan, P., Lin, C.S., Wang, G.: Schouten tensor and some topological properties. Commun. Anal. Geom. 13, 887–902 (2005)

    Article  MathSciNet  Google Scholar 

  33. Gonźalez, M.: Singular sets of a class of fully non-linear equations in conformal geometry. Ph.D thesis, Princeton University (2004)

  34. Gonźalez, M.: Singular sets of a class of locally conformally flat manifolds. Duke Math. J. 129, 551–572 (2005)

    Article  MathSciNet  Google Scholar 

  35. Gonźalez, M.: Removability of singularities for a class of fully non-linear equations. Calc. Var. Partial Differ. Equ. 27, 439–466 (2006)

    Article  MathSciNet  Google Scholar 

  36. Gursky, M., Viaclovsky, J.: A new variational characterization of three dimensional space forms. Invent. Math. 145, 251–278 (2001)

    Article  MathSciNet  Google Scholar 

  37. Gursky, M., Viaclovsky, J.: Fully nonlinear equations on Riemannian manifolds with negative curvature. Indiana Univ. Math. J. 52, 399–420 (2003)

    Article  MathSciNet  Google Scholar 

  38. Gursky, M., Viaclovsky, J.: A fully nonlinear equation on four-manifolds with positive scalar curvature. J. Differ. Geom. 63, 131–154 (2003)

    Article  MathSciNet  Google Scholar 

  39. Gursky, M., Viaclovsky, J.: Convexity and singularities of curvature equations in conformal geometry. Int. Math. Res. Not. 2006, 96890–96890 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Gursky, M., Viaclovsky, J.: Prescribing symmetric functions of the eigenvalues of the Ricci tensor. Ann. Math. 166, 475–531 (2007)

    Article  MathSciNet  Google Scholar 

  41. Gursky, M., Viaclovsky, J.: Volume comparison and the \(\sigma _{k}\)-Yamabe problem. Adv. Math. 187, 447–487 (2004)

    Article  MathSciNet  Google Scholar 

  42. Gursky, M., Streets, J.: A formal Riemannian structure on conformal classes and uniqueness for the \(\sigma _{2}\)-Yamabe problem.arXiv:1603.07005v1

  43. Guan, P., Viaclovsky, J., Wang, G.: Some properties of the Schouten tensor and applications to conformal geometry. Trans. Am. Math. Soc. 355, 925–933 (2003)

    Article  MathSciNet  Google Scholar 

  44. Han, Z.: Local pointwise estimates for solutions of the \(\sigma _{2}\) curvature equation on 4-manifolds. Int. Math. Res. Not. 79, 4269–4292 (2004)

    Article  MathSciNet  Google Scholar 

  45. Han, Z., Li, Y.Y., Teixeira, E.: Asymptotic behavior of solutions to the \(\sigma _{k}\)-Yamabe equation near isolated singularities. Invent math. 182, 635–684 (2010)

    Article  MathSciNet  Google Scholar 

  46. Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler–Einstein metrics with edge singularities with an appendix by C. Li and Y. Rubinstein. Ann. Math. 183, 95–176 (2016)

    Article  MathSciNet  Google Scholar 

  47. Korevaar, N., Mazzeo, R., Pacard, F., Schoen, R.: Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math. 135, 233–272 (1999)

    Article  MathSciNet  Google Scholar 

  48. Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations. Commun. Pure Appl. Math. 56, 1414–1464 (2003)

    MathSciNet  MATH  Google Scholar 

  49. Li, A., Li, Y.Y.: On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe. Acta Math. 195, 117–154 (2005)

    Article  MathSciNet  Google Scholar 

  50. Li, Y.Y.: Conformally invariant fully nonlinear elliptic equations and isolated singularities. J. Funct. Anal. 233, 380–425 (2006)

    Article  MathSciNet  Google Scholar 

  51. Li, Y.Y.: Local gradient estimates of solutions to some conformally invariant fully nonlinear equations. C. R. Math. Acad. Sci. Paris 343, 249–252 (2006)

    Article  MathSciNet  Google Scholar 

  52. Li, Y.Y., Nguyen, L.: A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound. J. Funct. Anal. 266, 3741–3771 (2014)

    Article  MathSciNet  Google Scholar 

  53. Luo, F., Tian, G.: Liouville equation and spherical convex polytopes. Proc. Am. Math. Soc. 116, 1119–1129 (1992)

    Article  MathSciNet  Google Scholar 

  54. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)

    Article  MathSciNet  Google Scholar 

  55. Sheng, W.-M., Trudinger, N., Wang, X.-J.: The Yamabe problem for higher order curvatures. J. Differ. Geom. 77, 515–553 (2007)

    Article  MathSciNet  Google Scholar 

  56. Silva Santos, A.: Solutions to the singular \(\sigma _{2}\)-Yamabe problem with isolated singularities. Indiana Univ. Math. J. 66, 741–790 (2017)

    Article  MathSciNet  Google Scholar 

  57. Tian, G.: Kähler-Einstein metrics on algebraic manifolds. In: Transcendental Methods in Algebraic Geometry (Cetraro 1994), Lecture Notes in Mathematics, pp. 143–185 (1646)

    Google Scholar 

  58. Tian, G.: K-stability and K\(\ddot{a}\)hler-Einstein metrics. Commun. Pure Appl. Math. 68, 1085–1156 (2015)

    Article  Google Scholar 

  59. Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324, 793–821 (1991)

    Article  MathSciNet  Google Scholar 

  60. Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci 22, 265–274 (1968)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  62. Viaclovsky, J.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101, 283–316 (2000)

    Article  MathSciNet  Google Scholar 

  63. Viaclovsky, J.: Some fully nonlinear equations in conformal geometry. In: Differential Equations and Mathematical Physics, Birmingham, AL, 1999. AMS/IP Studies in Advanced Mathematics, vol. 16, pp. 425–433. American Mathematical Society, Providence (2000)

  64. Viaclovsky, J.: Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. Commun. Anal. Geom. 10, 815–846 (2002)

    Article  MathSciNet  Google Scholar 

  65. Viaclovsky, J.: Conformally invariant Monge-Ampère partial differential equations: global solutions. Trans. Am. Math. Soc. 352, 4371–4379 (2000)

    Article  Google Scholar 

  66. Wang, X.-J.: A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry. Chin. Ann. Math. Ser. B 27, 169–178 (2006)

    Article  MathSciNet  Google Scholar 

  67. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242, USA

    Hao Fang

  2. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China

    Wei Wei

Authors
  1. Hao Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Fang.

Additional information

Communicated by A. Chang.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

H. F.’s works is partially supported by a Simons Foundation research collaboration grant. W. W.’s works is supported by China Scholarship Council.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fang, H., Wei, W. \(\sigma _{2}\) Yamabe problem on conic 4-spheres. Calc. Var. 58, 119 (2019). https://doi.org/10.1007/s00526-019-1551-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1551-y

Mathematics Subject Classification

Search

Navigation