2013, 20: 97-102. doi: 10.3934/era.2013.20.97

Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants

1. 

0287 Frist Center, Princeton University, Princeton, NJ 08544, United States

2. 

Penn State University Mathematics Department, 206 McAllister Building, University Park, PA 16802, United States

Received  September 2013 Published  November 2013

We use the Hofer norm to show that all Hamiltonian diffeomorphisms with compact support in $\mathbb{R}^{2n}$ that displace an open connected set with a nonzero Hofer-Zehnder capacity move a point farther than a capacity-dependent constant. In $\mathbb{R}^2$, this result is extended to all compactly supported area-preserving homeomorphisms. Next, using the spectral norm, we show the result holds for Hamiltonian diffeomorphisms on closed surfaces. We then show that all area-preserving homeomorphisms of $S^2$ and $\mathbb{RP}^2$ that displace the closure of an open connected set of fixed area move a point farther than an area-dependent constant.
Citation: Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
References:
[1]

S. Seyfaddini, The displaced disks problem via symplectic topology,, \arXiv{1307.5704}., (). 

[2]

H. Hofer, Estimates for the energy of a symplectic map, Comment. Math Helv., 68 (1993), 48-72. doi: 10.1007/BF02565809.

[3]

S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626.

[4]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.

[5]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.

[6]

Y. G. Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphism,, \arXiv{math/0601183v5}., (). 

[7]

S. Seyfaddini, $C^0$-limits of Hamiltonian flows and Oh-Schwarz spectral invariants,, \arXiv{1109.4123v2}., (). 

[8]

Y.-G. Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 130 (2005), 199-295.

[9]

M. Usher, The sharp energy-capacity inequality, Comm. Contemp. Math., 12 (2010), 457-473. doi: 10.1142/S0219199710003889.

[10]

S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Topol. Anal., 4 (2012), 481-498. doi: 10.1142/S1793525312500215.

show all references

References:
[1]

S. Seyfaddini, The displaced disks problem via symplectic topology,, \arXiv{1307.5704}., (). 

[2]

H. Hofer, Estimates for the energy of a symplectic map, Comment. Math Helv., 68 (1993), 48-72. doi: 10.1007/BF02565809.

[3]

S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626.

[4]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.

[5]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.

[6]

Y. G. Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphism,, \arXiv{math/0601183v5}., (). 

[7]

S. Seyfaddini, $C^0$-limits of Hamiltonian flows and Oh-Schwarz spectral invariants,, \arXiv{1109.4123v2}., (). 

[8]

Y.-G. Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 130 (2005), 199-295.

[9]

M. Usher, The sharp energy-capacity inequality, Comm. Contemp. Math., 12 (2010), 457-473. doi: 10.1142/S0219199710003889.

[10]

S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Topol. Anal., 4 (2012), 481-498. doi: 10.1142/S1793525312500215.

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