American Institue of Mathematical Sciences

2014, 21: 80-88. doi: 10.3934/era.2014.21.80

Compactly supported Hamiltonian loops with a non-zero Calabi invariant

 1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel

Received  November 2013 Revised  February 2014 Published  May 2014

We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds.
Citation: Asaf Kislev. Compactly supported Hamiltonian loops with a non-zero Calabi invariant. Electronic Research Announcements, 2014, 21: 80-88. doi: 10.3934/era.2014.21.80
References:
 [1] A. Cannas da Silva, Symplectic Toric Manifolds,, 2001. Available from: \url{http://www.math.ist.utl.pt/~acannas/Books/toric.pdf}., (). [2] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds,, Memoirs Amer. Math. Soc., 141 (1999). doi: 10.1090/memo/0672. [3] D. McDuff, Loops in the Hamiltonian group: A survey,, in Symplectic Topology and Measure Preserving Dynamical Systems, (2010), 127. doi: 10.1090/conm/512/10061. [4] D. McDuff and D. Salamon, Introduction to Symplectic Topology,, Second edition, (1998). [5] L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms,, Lectures in Mathematics ETH Zürich, (2001). doi: 10.1007/978-3-0348-8299-6. [6] L. Polterovich, Hamiltonian loops and Arnold's principle,, in Topics in Singularity Theory, (1997), 181. [7] S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds,, J. Top. Anal., 4 (2012), 481. doi: 10.1142/S1793525312500215.

show all references

References:
 [1] A. Cannas da Silva, Symplectic Toric Manifolds,, 2001. Available from: \url{http://www.math.ist.utl.pt/~acannas/Books/toric.pdf}., (). [2] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds,, Memoirs Amer. Math. Soc., 141 (1999). doi: 10.1090/memo/0672. [3] D. McDuff, Loops in the Hamiltonian group: A survey,, in Symplectic Topology and Measure Preserving Dynamical Systems, (2010), 127. doi: 10.1090/conm/512/10061. [4] D. McDuff and D. Salamon, Introduction to Symplectic Topology,, Second edition, (1998). [5] L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms,, Lectures in Mathematics ETH Zürich, (2001). doi: 10.1007/978-3-0348-8299-6. [6] L. Polterovich, Hamiltonian loops and Arnold's principle,, in Topics in Singularity Theory, (1997), 181. [7] S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds,, J. Top. Anal., 4 (2012), 481. doi: 10.1142/S1793525312500215.
 [1] Fiammetta Battaglia and Elisa Prato. Nonrational, nonsimple convex polytopes in symplectic geometry. Electronic Research Announcements, 2002, 8: 29-34. [2] Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009 [3] Chungen Liu, Qi Wang. Symmetrical symplectic capacity with applications. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2253-2270. doi: 10.3934/dcds.2012.32.2253 [4] P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1 [5] Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13 [6] Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211 [7] Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239 [8] Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291 [9] Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437 [10] Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291 [11] Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407 [12] Bernd Kawohl, Jiří Horák. On the geometry of the $p$-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [13] Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105 [14] Marie-Claude Arnaud. When are the invariant submanifolds of symplectic dynamics Lagrangian?. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1811-1827. doi: 10.3934/dcds.2014.34.1811 [15] Michael Khanevsky. Hofer's length spectrum of symplectic surfaces. Journal of Modern Dynamics, 2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219 [16] Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077 [17] Daniel Guan. Classification of compact homogeneous spaces with invariant symplectic structures. Electronic Research Announcements, 1997, 3: 52-54. [18] Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367 [19] Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205 [20] Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

2016 Impact Factor: 0.483