×

Conformal complete metrics with prescribed non-negative Gaussian curvature in \({\mathbb{R}}^ 2\). (English) Zbl 0592.53034

The paper deals with the existence of (complete) Riemannian metrics g on \({\mathbb{R}}^ 2\), conformal to the Euclidean metric \(g_ 0\) (by \(g=e^{2u}g_ 0)\) and possessing a prescribed Gaussian curvature \(k: {\mathbb{R}}^ 2\to {\mathbb{R}}\). This problem is intimately related to the nonlinear elliptic PDE \(\Delta u+k(x)e^{2u}=0\). It is always assumed that k is Hölder-continuous and that k(x) for large \(| x|\) is asymptotically comparable to some negative power of \(| x|\), in one sense or the other. For example, if k is positive somewhere and, asymptotically, \(| k(x)| \leq M/| x|^ b\) for some constants \(M>0\), \(b\geq 2\) then there is a complete metric g, for which an asymptotic rule like \[ (b-c)\cdot \ln | x| -2\tilde C\leq 2u(x)\leq (b-c)\cdot \ln | x| +2\tilde C \] holds. In case \(k\geq 0\) the condition \(b\geq 2\) may be relaxed to \(b>0\), provided some other assumptions are correspondingly sharpened. This is the most complicated case considered here. Several situations are exhibited where every solution of the elliptic PDE has an asymptotic behaviour like this. Finally, it is shown in the present frame that Cohn-Vossen’s inequality is necessary and sufficient for the metric g to become complete. The methods are mainly analytical in nature, including e.g. the Leray- Schauder fixed point theory and the capacity of planar sets.
Reviewer: R.Walter

MSC:

53C20 Global Riemannian geometry, including pinching
35J60 Nonlinear elliptic equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [Ah] Ahlfors, L.: Conformal invariants. Topics in geometric function theory, McGraw-Hill Series in Higher Mathematics (1973) · Zbl 0272.30012
[2] [A] Aviles, P.: Prescribing conformal complete metrics with given positive Gaussian curvature in ?2, Berkeley, California: Mathematical Sciences Research Institute, Preprint, (1983)
[3] [Ba] Bandle, C.: Isoperimetric inequalities for a nonlinear eigenvalue problem. Proc. Am. Math. Soc.56, 243-246 (1976) · Zbl 0326.35004 · doi:10.1090/S0002-9939-1976-0477402-7
[4] [B] Bleecker, D.: The Gauss-Bonnet inequality and almost-geodesic loops. Adv. Math.14, 183-193 (1974) · Zbl 0289.53039 · doi:10.1016/0001-8708(74)90029-2
[5] [C] Calabi, E.: On Ricci curvature and geodesics. Duke Math. J.34, 667-675 (1967) · Zbl 0153.51501 · doi:10.1215/S0012-7094-67-03469-2
[6] [Ca] Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Am. Math. Soc.5, 235-263 (1981) · Zbl 0481.58023 · doi:10.1090/S0273-0979-1981-14934-X
[7] [Fe] Federer, H.: Curvature measures.Trans. Am. Math. Soc.93, 418-491 (1959) · Zbl 0089.38402 · doi:10.1090/S0002-9947-1959-0110078-1
[8] [F] Finn, R.: On a class of conformal metrics with applications to differential geometry in the large. Comment. Math. Helv.40, 1-30 (1965) · Zbl 0192.27301 · doi:10.1007/BF02564362
[9] [G-T] Gilbarg, D., Trudinger, N.A.: Elliptic partial differential equations of second order. Berlin-Heidelberg-New York: Springer (1977) · Zbl 0361.35003
[10] [G-L] Gromov, M., Lawson, B.: Positive scalar curvature and the dirac operator on complete Riemannian manifolds. (To appear) · Zbl 0538.53047
[11] [H-S] Hewitt, E., Stromberg, K.: Real and abstract analysis, Berlin-Heidelberg-New York: Springer (1965) · Zbl 0137.03202
[12] [H] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv.32, 13-72 (1957) · Zbl 0080.15001 · doi:10.1007/BF02564570
[13] [H, 1] Huber, A.: On the isoperimetric inequality on surfaces of variable Gaussian curvature. Ann. Math.60 No. 2, 237-247 (1954) · Zbl 0056.15801 · doi:10.2307/1969630
[14] [J] Jones, F.: Rudiments of Riemann surfaces. Ric. Univ. Lect. Notes Math., No. 2, (1971) · Zbl 0229.30004
[15] [K-W, 1] Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math.99, No. 1, 14-47 (1974) · Zbl 0273.53034 · doi:10.2307/1971012
[16] [K-W, 2] Kazdan, J., Warner, F.: Curvature functions for open 2-manifolds. Ann. Math.99, 203-219 (1974) · Zbl 0278.53031 · doi:10.2307/1970898
[17] [M] McOwen, R.: On the equation ?u+ke 2u =f and prescribed negative curvature in ?2, (To appear) · Zbl 0568.35035
[18] [Mc] McOwen, R.: Conformal metrics in ?2 with prescribed Gaussian and positive total curvature (To appear: Indiana Univ. Math. J.)
[19] [N, 1] Ni, W.M.: On the elliptic equation ?u+ku (n+2)/(n?2)=0 its generalizations and applications in geometry. Indiana Univ. Math. J.4,493-532 (1982) · Zbl 0496.35036 · doi:10.1512/iumj.1982.31.31040
[20] [N, 2] Ni, W.M.: On the elliptic equation ?u+k(x)e 2u =0 and conformal metrics with prescribed Gaussian curvature. Invent. Math.66, 343-353 (1982) · Zbl 0487.35042 · doi:10.1007/BF01389399
[21] [O] Oleinik, O.A.: On the equation ?u+k(x)e u +0. Russ. Math.Surv.33, 243-244 (1978) · Zbl 0401.35051 · doi:10.1070/RM1978v033n02ABEH002424
[22] [P] Payne, L.E.: Isoperimetric inequalities and their applications. SIAM Rev. Vol.9, No. 3 (1967) · Zbl 0154.12602
[23] [Sa] Sattinger, D.H.: Conformal metrics in ?2 with prescribed curvatures. Indiana Univ. Math. J.22, 1-4 (1972) · Zbl 0236.53009 · doi:10.1512/iumj.1972.22.22001
[24] [S] Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math.111,247-302 (1964) · Zbl 0128.09101 · doi:10.1007/BF02391014
[25] [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
[26] [T] Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl.110, 353-372 (1976) · Zbl 0353.46018 · doi:10.1007/BF02418013
[27] [U] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math., Physics83, 11-29 (1982) · Zbl 0491.58032 · doi:10.1007/BF01947068
[28] [W] Weinberger, H.F.: Symmetrization in uniformly elliptic problems. Studies in Mathematical Analysis and Related Topics. Essays in honor of G. P?lya, Stanford, California: Stanford University Press, (1962), pp. 424-428
[29] [Y] Yau, S.T.: Isoperimetric inequalities and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super.8, 487-507 (1975) · Zbl 0325.53039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.