Lagrangian submanifolds and moment convexity

Authors:
Bernhard Krötz and Michael Otto

Journal:
Trans. Amer. Math. Soc. **358** (2006), 799-818

MSC (2000):
Primary 53D20, 22E15

Published electronically:
May 10, 2005

MathSciNet review:
2177041

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a Hamiltonian torus action on a compact connected symplectic manifold and its associated momentum map . For certain Lagrangian submanifolds we show that is convex. The submanifolds arise as the fixed point set of an involutive diffeomorphism which satisfies several compatibility conditions with the torus action, but which is in general not anti-symplectic. As an application we complete a symplectic proof of Kostant's non-linear convexity theorem.

**[1]**A. Alekseev, E. Meinrenken, and C. Woodward,*Linearization of Poisson actions and singular values of matrix products*, Ann. Inst. Fourier (Grenoble)**51**(2001), no. 6, 1691–1717 (English, with English and French summaries). MR**1871286****[2]**M. F. Atiyah,*Convexity and commuting Hamiltonians*, Bull. London Math. Soc.**14**(1982), no. 1, 1–15. MR**642416**, 10.1112/blms/14.1.1**[3]**C. De Concini and C. Procesi,*Quantum groups*, 𝐷-modules, representation theory, and quantum groups (Venice, 1992), Lecture Notes in Math., vol. 1565, Springer, Berlin, 1993, pp. 31–140. MR**1288995**, 10.1007/BFb0073466**[4]**J. J. Duistermaat,*Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution*, Trans. Amer. Math. Soc.**275**(1983), no. 1, 417–429. MR**678361**, 10.1090/S0002-9947-1983-0678361-2**[5]**V. Guillemin and S. Sternberg,*Convexity properties of the moment mapping*, Invent. Math.**67**(1982), no. 3, 491–513. MR**664117**, 10.1007/BF01398933**[6]**Victor Guillemin and Shlomo Sternberg,*Symplectic techniques in physics*, 2nd ed., Cambridge University Press, Cambridge, 1990. MR**1066693****[7]**Peter Heinzner and Alan Huckleberry,*Kählerian potentials and convexity properties of the moment map*, Invent. Math.**126**(1996), no. 1, 65–84. MR**1408556**, 10.1007/s002220050089**[8]**Joachim Hilgert and Karl-Hermann Neeb,*Poisson Lie groups and non-linear convexity theorems*, Math. Nachr.**191**(1998), 153–187. MR**1621294**, 10.1002/mana.19981910108**[9]**Joachim Hilgert, Karl-Hermann Neeb, and Werner Plank,*Symplectic convexity theorems and coadjoint orbits*, Compositio Math.**94**(1994), no. 2, 129–180. MR**1302314****[10]**Friedrich Knop,*Convexity of Hamiltonian manifolds*, J. Lie Theory**12**(2002), no. 2, 571–582. MR**1923787****[11]**Bertram Kostant,*On convexity, the Weyl group and the Iwasawa decomposition*, Ann. Sci. École Norm. Sup. (4)**6**(1973), 413–455 (1974). MR**0364552****[12]**Kurt Leichtweiss,*Konvexe Mengen*, Springer-Verlag, Berlin-New York, 1980 (German). Hochschultext. MR**586235****[13]**Jiang-Hua Lu and Tudor Ratiu,*On the nonlinear convexity theorem of Kostant*, J. Amer. Math. Soc.**4**(1991), no. 2, 349–363. MR**1086967**, 10.1090/S0894-0347-1991-1086967-2**[14]**Reyer Sjamaar,*Convexity properties of the moment mapping re-examined*, Adv. Math.**138**(1998), no. 1, 46–91. MR**1645052**, 10.1006/aima.1998.1739

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Additional Information

**Bernhard Krötz**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1221

Email:
kroetz@math.uoregon.edu

**Michael Otto**

Affiliation:
Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio

Email:
otto@math.ohio-state.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03723-2

Received by editor(s):
November 11, 2003

Received by editor(s) in revised form:
March 31, 2004

Published electronically:
May 10, 2005

Additional Notes:
The work of the first author was supported in part by NSF grant DMS-0097314

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.