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)
[Ba] Bandle, C.: Isoperimetric inequalities for a nonlinear eigenvalue problem. Proc. Am. Math. Soc.56, 243–246 (1976)
[B] Bleecker, D.: The Gauss-Bonnet inequality and almost-geodesic loops. Adv. Math.14, 183–193 (1974)
[C] Calabi, E.: On Ricci curvature and geodesics. Duke Math. J.34, 667–675 (1967)
[Ca] Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Am. Math. Soc.5, 235–263 (1981)
[Fe] Federer, H.: Curvature measures.Trans. Am. Math. Soc.93, 418–491 (1959)
[F] Finn, R.: On a class of conformal metrics with applications to differential geometry in the large. Comment. Math. Helv.40, 1–30 (1965)
[G-T] Gilbarg, D., Trudinger, N.A.: Elliptic partial differential equations of second order. Berlin-Heidelberg-New York: Springer (1977)
[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)
[H] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32, 13–72 (1957)
[H, 1] Huber, A.: On the isoperimetric inequality on surfaces of variable Gaussian curvature. Ann. Math.60 No. 2, 237–247 (1954)
[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)
[K-W, 2] Kazdan, J., Warner, F.: Curvature functions for open 2-manifolds. Ann. Math.99, 203–219 (1974)
[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)
[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)
[O] Oleinik, O.A.: On the equation Δu+k(x)e u+0. Russ. Math.Surv.33, 243–244 (1978)
[P] Payne, L.E.: Isoperimetric inequalities and their applications. SIAM Rev. Vol.9, No. 3 (1967)
[Sa] Sattinger, D.H.: Conformal metrics in ℝ2 with prescribed curvatures. Indiana Univ. Math. J.22, 1–4 (1972)
[S] Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math.111,247–302 (1964)
[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)
[U] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math., Physics83, 11–29 (1982)
[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
[Y] Yau, S.T.: Isoperimetric inequalities and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super.8, 487–507 (1975)
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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
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DOI: https://doi.org/10.1007/BF01394420