Harmonic diffeomorphisms between the annuli with rotational symmetry. (English) Zbl 1296.31001
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 101, 144-150 (2014); addendum ibid. 105, 1-2 (2014).
The authors deal with the existence or non-existence of rotationally symmetric harmonic diffeomorphisms between the annuli with the Poincaré metric, or Euclidean metric on the target manifold. There are many results on existence or non-existence of harmonic diffeomorphisms between Riemannian manifolds, or from the unit disk onto the plane. For instance, E. Heinz [Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., Math.-Phys.-Chem. Abt. 1952, 51–56 (1952; Zbl 0048.15401)] proved the non-existence of harmonic diffeomorphism from the unit disk onto \({\mathbb C}\) with its flat metric. Recently, P. Collin and H. Rosenberg [Ann. Math. (2) 172, No. 3, 1879–1906 (2010; Zbl 1209.53010)] constructed harmonic diffeomorphisms from \({\mathbb C}\) onto the Poincaré disk, and M. Leguil and H. Rosenberg [“On harmonic diffeomorphisms from conformal annuli to Riemannian annuli”, Preprint] showed the non-existence of harmonic diffeomorphism from the punctured disk \({\mathbb D}\setminus \{0\}\) onto the flat cylinder \({\mathbb S}^1 \times {\mathbb R}\).
Denote the annulus by \(P(a) = {\mathbb D} \setminus \{|z| \leq a\}\) for \(0 < a < 1\). The authors prove that for \(0 < a \leq b <1\), there is a rotationally symmetric harmonic diffeomorphism from \(P(b)\) onto \(P(a)\) with the Poincaré metric. In fact, the authors obtain some necessary and sufficient conditions for the existence of rotationally symmetric harmonic diffeomorphism, which are given by inequalities for \(a\) and \(b\). They also show that there is a rotationally symmetric harmonic diffeomorphism from \(P(b)\) onto \(P(a)\) with its Euclidean metric if and only if \(a \leq \frac{2b}{1+b^2}\), and there is no rotationally symmetric harmonic diffeomorphism from \(P(a)\) onto \({\mathbb C}\setminus \{0\}\) with its Euclidean metric if and only if \(a \leq \frac{2b}{1+b^2}\) and vice versa.
Denote the annulus by \(P(a) = {\mathbb D} \setminus \{|z| \leq a\}\) for \(0 < a < 1\). The authors prove that for \(0 < a \leq b <1\), there is a rotationally symmetric harmonic diffeomorphism from \(P(b)\) onto \(P(a)\) with the Poincaré metric. In fact, the authors obtain some necessary and sufficient conditions for the existence of rotationally symmetric harmonic diffeomorphism, which are given by inequalities for \(a\) and \(b\). They also show that there is a rotationally symmetric harmonic diffeomorphism from \(P(b)\) onto \(P(a)\) with its Euclidean metric if and only if \(a \leq \frac{2b}{1+b^2}\), and there is no rotationally symmetric harmonic diffeomorphism from \(P(a)\) onto \({\mathbb C}\setminus \{0\}\) with its Euclidean metric if and only if \(a \leq \frac{2b}{1+b^2}\) and vice versa.
Reviewer: Gabjin Yun (Yongin)
References:
| [1] | Akutagawa, K., Harmonic diffeomorphisms of the hyperbolic plane, Trans. Amer. Math. Soc., 342, 1, 325-342 (1994) · Zbl 0796.58014 |
| [2] | Akutagawa, K.; Nishikawa, S., The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space, Tohoku Math. J. (2), 42, 1, 67-82 (1990) · Zbl 0679.53002 |
| [3] | T. K.-K., Au; Wan, T. Y.-H., Images of harmonic maps with symmetry, Tohoku Math. J. (2), 57, 3, 321-333 (2005) · Zbl 1087.53059 |
| [4] | Au, T. K.-K.; Tam, L.-F.; Wan, T. Y.-H., Hopf differentials and the images of harmonic maps, Comm. Anal. Geom., 10, 3, 515-573 (2002) · Zbl 1040.58004 |
| [5] | Au, T. K.-K.; Wan, T. Y.-H., From local solutions to a global solution for the equation \(\Delta w = e^{2 w} - \| \Phi \|^2 e^{- 2 w}\), (Geometry and Global Analysis (Sendai, 1993) (1993), Tohoku Univ: Tohoku Univ Sendai), 427-430 · Zbl 0925.35057 |
| [7] | Chen, Q.; Eichhorn, J., Harmonic diffeomorphisms between complete Riemann surfaces of negative curvature, Asian J. Math., 13, 4, 473-533 (2009) · Zbl 1200.58010 |
| [8] | Cheung, L. F.; Law, C. K., An initial value approach to rotationally symmetric harmonic maps, J. Math. Anal. Appl., 289, 1, 1-13 (2004) · Zbl 1109.58021 |
| [9] | Choi, H. I.; Treibergs, A., New examples of harmonic diffeomorphisms of the hyperbolic plane onto itself, Manuscripta Math., 62, 2, 249-256 (1988) · Zbl 0669.58012 |
| [10] | Choi, H. I.; Treibergs, A., Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space, J. Differential Geom., 32, 3, 775-817 (1990) · Zbl 0717.53038 |
| [11] | Collin, P.; Rosenberg, H., Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math. (2), 172, 3, 1879-1906 (2010) · Zbl 1209.53010 |
| [12] | Coron, J.-M.; Hélein, F., Harmonic diffeomorphisms, minimizing harmonic maps and rotational symmetry, Compos. Math., 69, 2, 175-228 (1989) · Zbl 0686.58012 |
| [13] | Gálvez, J. A.; Rosenberg, H., Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces, Amer. J. Math., 132, 5, 1249-1273 (2010) · Zbl 1229.53064 |
| [14] | Han, Z.-C., Remarks on the geometric behavior of harmonic maps between surfaces, (Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994) (1996), A K Peters: A K Peters Wellesley, MA), 57-66 · Zbl 0871.58025 |
| [15] | Han, Z.-C.; Tam, L.-F.; Treibergs, A.; Wan, T. Y.-H., Harmonic maps from the complex plane into surfaces with nonpositive curvature, Comm. Anal. Geom., 3, 1-2, 85-114 (1995) · Zbl 0843.58028 |
| [16] | Hélein, F., Harmonic diffeomorphisms between Riemannian manifolds, (Variational Methods (Paris, 1988). Variational Methods (Paris, 1988), Progr. Nonlinear Differential Equations Appl., vol. 4 (1990), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 309-318 · Zbl 0738.58015 |
| [17] | Hélein, F., Harmonic diffeomorphisms with rotational symmetry, J. Reine Angew. Math., 414, 45-49 (1991) · Zbl 0726.58017 |
| [18] | Heinz, E., Über die Lüsungen der Minimalflächengleichung, (Nachr. Akad. Wiss. Göttingen. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. Math.-Phys.-Chem. Abt. (1952)), 51-56, (in German) · Zbl 0048.15401 |
| [19] | Jost, J.; Schoen, R., On the existence of harmonic diffeomorphisms, Invent. Math., 66, 2, 353-359 (1982) · Zbl 0488.58009 |
| [20] | Kalaj, D., On harmonic diffeomorphisms of the unit disc onto a convex domain, Complex Var. Theory Appl., 48, 2, 175-187 (2003) · Zbl 1041.30006 |
| [22] | Li, P.; Tam, L.-F.; Wang, J.-P., Harmonic diffeomorphisms between Hadamard manifolds, Trans. Amer. Math. Soc., 347, 9, 3645-3658 (1995) · Zbl 0855.58020 |
| [23] | Lohkamp, J., Harmonic diffeomorphisms and Teichmüller theory, Manuscripta Math., 71, 4, 339-360 (1991) · Zbl 0734.30038 |
| [24] | Markovic, V., Harmonic diffeomorphisms and conformal distortion of Riemann surfaces, Comm. Anal. Geom., 10, 4, 847-876 (2002) · Zbl 1024.53044 |
| [25] | Markovic, V., Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces, J. Lond. Math. Soc., 65, 1, 103-114 (2002) · Zbl 1041.30020 |
| [26] | Ratto, A.; Rigoli, M., On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Differential Equations, 101, 1, 15-27 (1993) · Zbl 0767.34029 |
| [27] | Schoen, R. M., The role of harmonic mappings in rigidity and deformation problems, (Complex Geometry (Osaka, 1990). Complex Geometry (Osaka, 1990), Lecture Notes in Pure and Appl. Math., vol. 143 (1993), Dekker: Dekker New York), 179-200 · Zbl 0806.58013 |
| [28] | Shi, Y., On the construction of some new harmonic maps from \(R^m\) to \(H^m\), Acta Math. Sin. (Eng. Ser.), 17, 2, 301-304 (2001) · Zbl 0989.58004 |
| [29] | Shi, Y.; Tam, L.-F., Harmonic maps from \(R^n\) to \(H^m\) with symmetry, Pacific J. Math., 202, 1, 227-256 (2002) · Zbl 1055.58007 |
| [30] | Tachikawa, A., Rotationally symmetric harmonic maps from a ball into a warped product manifold, Manuscripta Math., 53, 3, 235-254 (1985) · Zbl 0578.58008 |
| [31] | Tachikawa, A., A nonexistence result for harmonic mappings from \(R^n\) into \(H^n\), Tokyo J. Math., 11, 2, 311-316 (1988) · Zbl 0686.58013 |
| [32] | Tam, L.-F.; Wan, T. Y.-H., Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials, Comm. Anal. Geom., 2, 4, 593-625 (1994) · Zbl 0842.58014 |
| [33] | Tam, L.-F.; Wan, T. Y.-H., Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space, J. Differential Geom., 42, 2, 368-410 (1995) · Zbl 0873.32019 |
| [34] | Wan, T. Y.-H., Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Differential Geom., 35, 3, 643-657 (1992) · Zbl 0808.53056 |
| [35] | Wan, Tom Y. H., Review on harmonic diffeomorphisms between complete noncompact surfaces, (Surveys in Geometric Analysis and Relativity. Surveys in Geometric Analysis and Relativity, Adv. Lect. Math. (ALM), vol. 20 (2011), Int. Press: Int. Press Somerville, MA), 509-516 · Zbl 1268.53069 |
| [36] | Wu, D., Harmonic diffeomorphisms between complete surfaces, Ann. Global Anal. Geom., 15, 2, 133-139 (1997) · Zbl 0884.58033 |
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.
