Abstract
For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kähler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce \(Z(M, [[\omega ]])\) depending on symplectic deformation equivalence class \([[\omega ]]\). We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of \(Z(M, [[\omega ]] )\). Using Z invariants, we set up a Kazdan–Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold \((M, \omega )\) of dimension \(\ge \!\!4\), any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to \(\omega \).
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Notes
For example, one can apply Protter’s theorem [20] which states that if a real-valued function u defined in a domain \(D\subset \mathbb R^m\) containing 0 satisfies that \(|\Delta ^nu|\le f(x,u,Du,\ldots ,D^ku)\) for Lipschitzian f and \(k\le [\frac{3n}{2}]\), and \(e^{2r^{-\beta }}u\rightarrow 0\) as \(r:=\sqrt{x_1^2+\cdots +x_m^2}\rightarrow 0\) for any constant \(\beta >0\), then u vanishes identically in D.
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Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MOE), for the first author (NRF-2010-0011704)..
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Kim, J., Sung, C. Scalar Curvature Functions of Almost-Kähler Metrics. J Geom Anal 26, 2711–2728 (2016). https://doi.org/10.1007/s12220-015-9645-z
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DOI: https://doi.org/10.1007/s12220-015-9645-z