Vanishing Pohozaev constant and removability of singularities
Jürgen Jost, Chunqin Zhou, and Miaomiao Zhu
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Submission date: 19. Nov. 2015
published in: Journal of differential geometry, 111 (2019) 1, p. 91-144
DOI number (of the published article): 10.4310/jdg/1547607688
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Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the validity of the Pohozaev identity. In situations where such an identity fails to hold, we introduce a new quantity, called the Pohozaev constant, which on one hand measures the extent to which the Pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity. We apply this to the blow-up analysis for super-Liouville type equations on Riemann surfaces with conical singularities, because in the presence of such singularities, conformal invariance no longer holds and a local singularity is in general non-removable unless the Pohozaev constant is vanishing.