Abstract
A Grünbaum type of measure of symmetry is calculated and estimated for the DoCarmo-Wallach moduli spaces for eigenmaps and spherical minimal immersions. The DeTurck-Ziller classification of minimal imbeddings of 3-dimensional space forms is used to obtain exact determination of the measure for the SU(2)-equivariant moduli.
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References
Berger M.: Geometry I-II. Springer, Berlin (1987)
Besse A.: Manifolds All of Whose Geodesics are Closed. Springer, Berlin (1978)
Börner H.: Representations of Groups. North-Holland, Amsterdam (1963)
Calabi E.: Minimal immersions of surfaces in euclidean spheres. J. Differ. Geom. 1, 111–125 (1967)
DeTurck D., Ziller W.: Minimal isometric immersions of spherical space forms in spheres. J. Differ. Geom. 1, 111–125 (1967)
DeTurck D., Ziller W.: Minimal isometric immersions of spherical space forms into spheres. Comment. Math. Helvetici 67, 428–458 (1992)
DeTurck D., Ziller W.: Spherical minimal mmersions of spherical space forms. Proc. Symp. Pure Math. 54(Part I), 111–120 (1993)
DoCarmo M., Wallach N.: Minimal immersions of spheres into spheres. Ann. Math. 93, 43–62 (1971)
Eells J., Lemaire L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)
Escher Ch., Weingart G.: Orbits of SU(2)-representations and minimal isometric immersions. Math. Ann. 316, 743–769 (2000)
Grünbaum B.: Convex Polytopes. Springer, (2003)
Grünbaum B.: Measures of symmetry for convex sets. Proc. Symp. Pure Math. VII, 233–270 (1963)
Helgason, S.: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, vol. 83. American Mathematical Society Providence (2000)
Helgason, S.: Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39. American Mathematical Society Providence (1994)
Helgason S.: The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113, 153–180 (1965)
Knapp A.W.: Representation Theory of Semisimple Groups. Princeton University Press, Princeton, NJ (1986)
Mashimo K.: Minimal immersions of 3-dimensional spheres into spheres. Osaka J. Math. 2, 721–732 (1984)
Moore J.D.: Isometric immersions of space forms into space forms. Pacific J. Math. 40, 157–166 (1972)
Toth G.: On the structure of convex sets with applications to the moduli of spherical minimal immersions. Beitr. Algebra Geom. 49(2), 491–515 (2008)
Toth G.: On the shape of the moduli of spherical minimal immersions. Trans. Amer. Math. Soc. 358(6), 2425–2446 (2006)
Toth G.: Simplicial intersections of a convex set and moduli for spherical minimal immersions. Michigan Math. J. 52, 341–359 (2004)
Toth G.: Moduli for spherical maps and spherical minimal immersions of homogeneous spaces. J. Lie Theory 12, 551–570 (2002)
Toth G.: Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli. Springer, Berlin (2002)
Toth G.: Infinitesimal rotations of isometric minimal immersions between spheres. Amer. J. Math. 122, 117–152 (2000)
Toth G.: Eigenmaps and the space of minimal immersions between spheres. Indiana Univ. Math. J. 46(2), 637–658 (1997)
Toth G., Ziller W.: Spherical minimal immersions of the 3-sphere. Comment. Math. Helv. 74, 84–117 (1999)
Wallach, N.: Minimal immersions of symmetric spaces into spheres. In: Symmetric Spaces, Dekker, New York, pp. 1–40 (1972)
Weingart, G.: Geometrie der Modulräume minimaler isometrischer Immersionen der Sphären in Sphären, Bonner mathematische Schriften, Nr. 314, Bonn (1999)
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Toth, G. A measure of symmetry for the moduli of spherical minimal immersions. Geom Dedicata 160, 1–14 (2012). https://doi.org/10.1007/s10711-011-9667-z
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DOI: https://doi.org/10.1007/s10711-011-9667-z