Existence of Solutions to a super-Liouville equation with Boundary Conditions

TL;DR

This paper proves the existence of solutions to a super-Liouville equation with boundary conditions on a compact Riemannian surface with negative Euler characteristic, using variational methods and spectral analysis of a weighted Dirac operator.

Contribution

It introduces a weighted Dirac operator and constructs a Nehari manifold to establish solution existence for a complex super-Liouville equation with boundary conditions.

Findings

Existence of non-trivial solutions established.

Development of a spectral decomposition approach.

Application of minimax theory on the Nehari manifold.

Abstract

In this paper, we study the existence of solutions to a type of super-Liouville equation on the compact Riemannian surface $M$ with boundary and with its Euler characteristic $\chi(M)<0$. The boundary condition couples a Neumann condition for functions and a chirality boundary condition for spinors. Due to the generality of the equation, we introduce a weighted Dirac operator based on the solution to a related Liouville equation. Then we construct a Nehari manifold according to the spectral decomposition of the weighted Dirac operator, and use minimax theory on this Nehari manifold to show the existence of the non-trivial solutions.