Vanishing Pohozaev constant and removability of singularities

TL;DR

This paper establishes that the vanishing of the Pohozaev constant is both necessary and sufficient for the removability of singularities in certain conformally invariant problems, extending blow-up analysis techniques.

Contribution

It introduces the Pohozaev constant as a new tool to characterize singularities and applies it to super-Liouville equations on Riemann surfaces with conical singularities.

Findings

Pohozaev identity's validity characterizes singularity removability.

Vanishing Pohozaev constant indicates a removable singularity.

Application to super-Liouville equations with conical singularities.

Abstract

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 {\it 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.