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Title:
A Liouville-type theorem for certain degenerate-elliptic equations, and a classification of toric KALE/KALF scalar flat instantons
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Abstract:
It is known that Kahler reduction of scalar-flat toric surfaces leads to a pair of linear degenerate-elliptic equations x(fxx + fyy) − fx = 0 on the polytope. Vice-versa, solving these equations on the polytope give scalar-flat 4-manifolds. To the end of finding all admissible solutions, we study global aspects of the linear degenerate-elliptic equations of this form. Such PDE have been studied locally, but we obtain a new global classification of solutions, which amounts to a Liouville-type theorem. This provides a classification of scalar-flat metrics under certain natural conditions.
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