Energy quantization for a singular super-Liouville boundary value problem

TL;DR

This paper establishes energy quantization for super-Liouville equations on Riemann surfaces with boundary singularities, introducing a new method based on Pohozaev constant vanishing due to the loss of conformal invariance.

Contribution

It develops a novel approach using Pohozaev constant vanishing to analyze boundary singularities in super-Liouville equations, overcoming the challenge posed by conical singularities.

Findings

Proves energy quantization for solutions with boundary singularities.

Introduces a new method based on Pohozaev constant for boundary regularity.

Shows removability of boundary singularities under certain conditions.

Abstract

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.