Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-28T07:43:29.166Z Has data issue: false hasContentIssue false
  • English
  • Français

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

Published online by Cambridge University Press:  01 November 2008

DANIELA KRAUS  and
OLIVER ROTH
DANIELA KRAUS
Affiliation:
Universität Würzburg, Mathematisches Institut, D–97074 Würzburg, Germany. e-mail: dakraus@mathematik.uni-wuerzburg.de, roth@mathematik.uni-wuerzburg.de
OLIVER ROTH
Affiliation:
Universität Würzburg, Mathematisches Institut, D–97074 Würzburg, Germany. e-mail: dakraus@mathematik.uni-wuerzburg.de, roth@mathematik.uni-wuerzburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu = 4e2u near isolated singularities is generalized to solutions of the Gaussian curvature equation Δu = −κ(z)e2u where κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 643 - 667
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ahlfors, L.An extension of Schwarz's lemma. Trans. Amer. Math. Soc. 42 (1938), 359–364.Google Scholar
[2]Aviles, P. and McOwen, R.Conformal deformations of complete manifolds with negative curvature. J. Diff. Geom. 21 (1985), 269–281.Google Scholar
[3]Bieberbach, L. Δ u = e u und die automorphen Funktionen. Nachr. Akad. Wiss. Goett. II. Math. Phys. Kl. (1912), 599–602.Google Scholar
[4]Bieberbach, L.Δ u = e u und die automorphen Funktionen. Math. Ann. 77 (1977), 173–212.CrossRefGoogle Scholar
[5]Bland, J. S.Local boundary regularity of the canonical Einstein–Kähler metric on pseudo-convex domains. Math. Ann. 263 (1983), 289–301.CrossRefGoogle Scholar
[6]Chang, S.-Y. A.Non-linear elliptic equations in conformal geometry. European Mathematical Society, (2004); (Zurich lectures in advanced mathematics).CrossRefGoogle Scholar
[7]Chou, K. S. and Wan, T.Asymptotic radial symmetry for solutions of Δ u + e u = 0 in a punctured disc. Pacific J. Math. 163 (1994), 269–276.CrossRefGoogle Scholar
[8]Chou, K. S. and Wan, T. Correction to “Asymptotic radial symmetry for solutions of Δ u + e u = 0 in a punctured disc”. Pacific J. Math. 171 (1995), 589–590.CrossRefGoogle Scholar
[9]Elstrodt, J.Maβ– und Integrationstheorie (Springer, 1991).Google Scholar
[10]Gilbarg, D. and Trudinger, N. S.Elliptic Partial Differential Equations of Second Order (Springer, 1997).Google Scholar
[11]Gill, B. T. and Macgregor, T.Derivatives of the hyperbolic density near an isolated boundary point. Rocky Mountain J. Math. 36 (2006), 1873–1884.CrossRefGoogle Scholar
[12]Hayman, W. K. and Kennedy, P. B.Subharmonic Functions, Vol. 1 (Academic Press, 1976).Google Scholar
[13]Heins, M.On a class of conformal metrics. Nagoya Math. J. 21 (1962), 1–60.CrossRefGoogle Scholar
[14]Huber, A.Vollständige konforme Metriken und isolierte Singularitäten subharmonischer Funktionen. Comment. Math. Helv. 41 (1966), 105–136.CrossRefGoogle Scholar
[15]Hulin, D. and Troyanov, M.Prescribing curvature on open surfaces. Math. Ann. 293 (1992), 277–315.CrossRefGoogle Scholar
[16]Jost, J.Partial Differential Equations (Springer, 2002).Google Scholar
[17]Kraus, D., Roth, O. and Ruscheweyh, St.A boundary version of Ahlfors' Lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps. J. Anal. Math. 101 (2007), 219–256.CrossRefGoogle Scholar
[18]Liouville, J.Sur l'équation aux différences partielles . J. de Math. 16 (1853), 71–72.Google Scholar
[19]Mazzeo, R. and Taylor, M.Curvature and uniformization. Israel J. Math. 130 (2002), 323–346.CrossRefGoogle Scholar
[20]McOwen, R. C.Prescribed curvature and singularities of conformal metrics on Riemann surfaces. J. Math. Anal. Appl. 177 no. 1 (1993), 287–298.CrossRefGoogle Scholar
[21]Minda, D.The density of the hyperbolic metric near an isolated boundary point. Complex Variables 32 (1997), 331–340.Google Scholar
[22]Nitsche, J.Über die isolierten Singularitäten der Lösungen von Δ u = e u. Math. Z. 68 (1957), 316–324.CrossRefGoogle Scholar
[23]Picard, E.De l'équation Δ u = e u sur une surface de Riemann fermée. J. de Math. 9, no. 4 (1893), 273–291.Google Scholar
[24]Picard, E.De l'integration de l'équation differentielles Δ u = e u sur une surface de Riemann fermée. J. Reine Angew. Math. 130 (1905), 243–258.CrossRefGoogle Scholar
[25]Poincaré, H.Les fonctions fuchsiennes et l'équation Δ u = e u. J. de Math. 4, no. 5 (1898), 137–230.Google Scholar
[26]P'olya, G. and Szegö, G.Problems and Theorems in Analysis I (Springer-Verlag, 1976).CrossRefGoogle Scholar
[27]Rad'o, T.Subharmonic functions. Ergebnisse der Mathematik und ihrer Grenzgebiete (reprinted by Chelsea, 1949).Google Scholar
[28]Ransford, T. J.Potential Theory in the Complex Plane (Cambridge University Press, 1995).CrossRefGoogle Scholar
[29]Schippers, E.Conformal invariants and higher-order Schwarz lemmas. J. Anal. Math. 90 (2003), 217–241.CrossRefGoogle Scholar
[30]Schwarz, H. A. Preisaufgabe der Math.–Phys. Klasse der Königl. Ges. der Wissen-schaften zu Göttingen für das Jahr 1891. Nachr. Akad. Wiss. Gött. (1890), 216.Google Scholar
[31]Troyanov, M.Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 no. 2 (1990), 793–821.CrossRefGoogle Scholar
[32]Troyanov, M.The Schwarz lemma for nonpositively curved Riemannian surfaces. Manuscr. Math. 72 no. 3 (1991), 251–256.CrossRefGoogle Scholar
[33]Yamada, A.Bounded analytic functions and metrics of constant curvature on Riemann surfaces. Kodai Math. J. 11, no. 3 (1988), 317–324.CrossRefGoogle Scholar
[34]Yau, S. T.A general Schwarz Lemma for Kähler Manifolds. Amer. J. Math. 100 (1978), 197–203.CrossRefGoogle Scholar
[35]Yunyan, Y.Local estimates of singular solution to Gaussian curvature equation. J. Partial Differential Equations 16 (2003), 169–185.Google Scholar