Skip to main content
Log in

Conformal complete metrics with prescribed non-negative Gaussian curvature in ℝ2

  • Published:
Inventiones mathematicae Aims and scope

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

References

  • [Ah] Ahlfors, L.: Conformal invariants. Topics in geometric function theory, McGraw-Hill Series in Higher Mathematics (1973)

  • [A] Aviles, P.: Prescribing conformal complete metrics with given positive Gaussian curvature in ℝ2, Berkeley, California: Mathematical Sciences Research Institute, Preprint, (1983)

    Google Scholar 

  • [Ba] Bandle, C.: Isoperimetric inequalities for a nonlinear eigenvalue problem. Proc. Am. Math. Soc.56, 243–246 (1976)

    Google Scholar 

  • [B] Bleecker, D.: The Gauss-Bonnet inequality and almost-geodesic loops. Adv. Math.14, 183–193 (1974)

    Google Scholar 

  • [C] Calabi, E.: On Ricci curvature and geodesics. Duke Math. J.34, 667–675 (1967)

    Google Scholar 

  • [Ca] Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Am. Math. Soc.5, 235–263 (1981)

    Google Scholar 

  • [Fe] Federer, H.: Curvature measures.Trans. Am. Math. Soc.93, 418–491 (1959)

    Google Scholar 

  • [F] Finn, R.: On a class of conformal metrics with applications to differential geometry in the large. Comment. Math. Helv.40, 1–30 (1965)

    Google Scholar 

  • [G-T] Gilbarg, D., Trudinger, N.A.: Elliptic partial differential equations of second order. Berlin-Heidelberg-New York: Springer (1977)

    Google Scholar 

  • [G-L] Gromov, M., Lawson, B.: Positive scalar curvature and the dirac operator on complete Riemannian manifolds. (To appear)

  • [H-S] Hewitt, E., Stromberg, K.: Real and abstract analysis, Berlin-Heidelberg-New York: Springer (1965)

    Google Scholar 

  • [H] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32, 13–72 (1957)

    Google Scholar 

  • [H, 1] Huber, A.: On the isoperimetric inequality on surfaces of variable Gaussian curvature. Ann. Math.60 No. 2, 237–247 (1954)

    Google Scholar 

  • [J] Jones, F.: Rudiments of Riemann surfaces. Ric. Univ. Lect. Notes Math., No. 2, (1971)

  • [K-W, 1] Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math.99, No. 1, 14–47 (1974)

    Google Scholar 

  • [K-W, 2] Kazdan, J., Warner, F.: Curvature functions for open 2-manifolds. Ann. Math.99, 203–219 (1974)

    Google Scholar 

  • [M] McOwen, R.: On the equation Δu+ke 2u=f and prescribed negative curvature in ℝ2, (To appear)

  • [Mc] McOwen, R.: Conformal metrics in ℝ2 with prescribed Gaussian and positive total curvature (To appear: Indiana Univ. Math. J.)

  • [N, 1] Ni, W.M.: On the elliptic equation Δu+ku (n+2)/(n−2)=0 its generalizations and applications in geometry. Indiana Univ. Math. J.4,493–532 (1982)

    Google Scholar 

  • [N, 2] Ni, W.M.: On the elliptic equation Δu+k(x)e 2u=0 and conformal metrics with prescribed Gaussian curvature. Invent. Math.66, 343–353 (1982)

    Google Scholar 

  • [O] Oleinik, O.A.: On the equation Δu+k(x)e u+0. Russ. Math.Surv.33, 243–244 (1978)

    Google Scholar 

  • [P] Payne, L.E.: Isoperimetric inequalities and their applications. SIAM Rev. Vol.9, No. 3 (1967)

    Google Scholar 

  • [Sa] Sattinger, D.H.: Conformal metrics in ℝ2 with prescribed curvatures. Indiana Univ. Math. J.22, 1–4 (1972)

    Google Scholar 

  • [S] Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math.111,247–302 (1964)

    Google Scholar 

  • [S-Y] Schoen, R., Yau, S.T.: Complete three dimensional manifolds with positive Ricci curvature and scalar curvature. In: S.T. Yau (ed.) Seminar on Differential Geometric, Ann. Math. Stud. Princeton University Press(1982) pp. 209–228

  • [T] Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl.110, 353–372 (1976)

    Google Scholar 

  • [U] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math., Physics83, 11–29 (1982)

    Google Scholar 

  • [W] Weinberger, H.F.: Symmetrization in uniformly elliptic problems. Studies in Mathematical Analysis and Related Topics. Essays in honor of G. Pólya, Stanford, California: Stanford University Press, (1962), pp. 424–428

    Google Scholar 

  • [Y] Yau, S.T.: Isoperimetric inequalities and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super.8, 487–507 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aviles, P. Conformal complete metrics with prescribed non-negative Gaussian curvature in ℝ2 . Invent Math 83, 519–544 (1986). https://doi.org/10.1007/BF01394420

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01394420

Keywords

Navigation