Convergence of solutions of H-systems or how to blow bubbles. (English) Zbl 0584.49024

The authors study the dependence of parametric surfaces of constant mean curvature on the boundary curve when the boundary curve degenerates to one point. They look at the normalized situation where the surface \(u=(u_ 1,u_ 2,u_ 3)\), \(u=u(x,y)\) is parametrized on the unit disk B in \({\mathbb{R}}^ 2\) and possesses mean curvature 1. The surface is given in conformal parameters and the boundary condition is of Plateau type, i.e. u maps \(\partial B\) onto some Jordan curve \(\Gamma\) in \({\mathbb{R}}^ 3\). Now let \(\Gamma_ n\) be a sequence of Jordan curves with \(\Gamma_ n\to 0\) (n\(\to \infty)\) and let \(u^ n\) be surfaces as above, bounded by \(\Gamma_ n\), with uniformly bounded area. Then it is proved that a subsequence of \(u_ n\) converges to 0 or to a finite union of spheres of radius one. If one takes as \(u^ n\) large solutions of the problem then a subsequence converges to a single sphere of radius one containing 0.
The proofs rely on a blow up argument which leads to the system \(\Delta w=2w_ x\bigwedge w_ y\) on \({\mathbb{R}}^ 2\) with finite Dirichlet integral. A careful analysis shows that w is the stereographic projection of a rational function on \({\mathbb{C}}\). Dirichlet’s integral is in \(8\pi {\mathbb{Z}}^+.\)
The main result is that any sequence of solutions of the Dirichlet problem \(\Delta u^ n=2u^ n_ x\bigwedge u^ n_ y\) on B, \(u^ n=\gamma^ n\) on \(\partial B\) with \(\gamma^ n\to 0\) (n\(\to \infty)\), bounded in \(H^{1,2}(B,{\mathbb{R}}^ 3)\) after a blow up essentially behaves like a finite superposition of solutions w as above.
Reviewer: G.Dziuk


49Q05 Minimal surfaces and optimization
35J60 Nonlinear elliptic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI


[1] V. Benci & J. M. Coron, The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere. Ann. I.H.P. Analyse Nonlinéaire (to appear). · Zbl 0597.35022
[2] H. Brezis & J. M. Coron, Multiple solutions of H-systems and Rellich’s conjecture. Comm. Pure Appl. Math. 37 (1984), 149-187. See also Sur la conjecture de Rellich pour les surfaces à courbure moyenne prescrite. C. R. Acad. Sc. Paris 295 (1982), 615-618. · Zbl 0537.49022
[3] H. Brezis & J. M. Coron, Convergence de solutions de H-systèmes et application aux surfaces a courbure moyenne constante. C. R. Acad. Sc. Paris 298 (1984), 389-392.
[4] H. Brezis & E. H. Lieb, Minimum action solutions of some vector field equations. Comm. Math. Phys. (to appear).
[5] B. Gidas, W. M. Ni & L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 200-243. · Zbl 0425.35020
[6] R. D. Gulliver, R. Osserman & H. L. Royden, A theory of branched immersions of surfaces. Amer. J. Math. 95 (1973), 750-812. · Zbl 0295.53002
[7] E. Heinz, Über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung. Math. Ann. 127 (1954), 258-287. · Zbl 0055.15303
[8] S. Hildebrandt, On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23 (1970), 97-114. · Zbl 0181.38703
[9] S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings. Proc. 1980 Beijing Symp. Diff. Geom. and Diff. Eq., S. S. Chern and Wu Wen-tsün ed. Science Press Beijing (1982) and Gordon-Breach. · Zbl 0515.58012
[10] J. Jost, Harmonic mappings between surfaces. Lectures notes in Math. Volume 1062, Springer (1984).
[11] L. Lemaire, Applications harmoniques de surfaces Riemanniennes. J. Diff. Geometry 13 (1978), 51-78. · Zbl 0388.58003
[12] E. H. Lieb, On the lowest eigenvalue of the laplacian for the intersection of two domains. Invent. Math. 74 (1983), 441-448. · Zbl 0538.35058
[13] P. L. Lions, Principe de concentration-compacité en calcul des variations. C. R. Acad. Sci. Paris 294 (1982), 261-264. The concentration-compactness principle in the calculus of variations: the locally compact case. Part I, Ann. I.H.P. Analyse Non-linéaire 1 (1984), 109-145 and Part II 1 (1984), 223-283.
[14] P. L. Lions, Applications de la méthode de concentration compacité à l’existence de fonctions extrémales. C. R. Acad. Sci. Paris 296 (1983) p. 645-648. The concentration-compactness principle in the calculus of variations: the limit case. Parts I and II, Riv. Iberoamericana (to appear).
[15] W. Meeks & S. T. Yau, Topology of three dimensional manifolds and the embedding problems in minimal surface theory. Annals of Math. 112 (1980), 441-484. · Zbl 0458.57007
[16] E. A. Ruh, Asymptotic behaviour of non-parametric minimal hypersurfaces. J. Diff. Geometry, 4 (1970), 509-513. · Zbl 0206.50203
[17] J. Sacks & K. Uhlenbeck, The existence of minimal immersions of 2-spheres. Ann. Math. 113 (1981), 1-24. · Zbl 0462.58014
[18] J. Serrin, On surfaces of constant mean curvature which span a given space curve. Math. Z. 112 (1969), 77-88. · Zbl 0182.24001
[19] Y.-T. Siu & S. T. Yau. Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), 189-204. · Zbl 0442.53056
[20] G. Springer, Introduction to Riemann surfaces. Addison-Wesley, Reading MA-London (1957). · Zbl 0078.06602
[21] K. Steffen, Flächen konstanter mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Archive Rational Mech. Anal. 49 (1972), 99-128. · Zbl 0259.53043
[22] K. Steffen, On the nonuniqueness of surfaces with prescribed mean curvature spanning a given contour, (to appear). · Zbl 0678.49036
[23] M. Struwe, Non uniqueness in the Plateau problem for surfaces of constant mean curvature, (to appear). · Zbl 0603.49027
[24] M. Struwe, A global existence result for elliptic boundary value problems involving limiting nonlinearities, (to appear). · Zbl 0535.35025
[25] C. H. Taubes, Path connected Yang-Mills moduli spaces. J. Diff. Geom. (to appear). · Zbl 0551.53040
[26] H. Wente, An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26 (1969), 318-344. · Zbl 0181.11501
[27] H. Wente, The differential equation 56-01 with vanishing boundary values. Proc. A.M.S. 50 (1975), 131-137. · Zbl 0313.35030
[28] H. Wente, Large solutions to the volume constrainted Plateau proplem. Arch. Rational Mech. Anal. 75 (1980), 59-77. · Zbl 0473.49029
[29] H. Werner, Das Problem von Douglas für Flächen konstanter mittlerer Krümmung. Math. Ann. 133 (1957), 303-319. · Zbl 0077.34901
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.