Super-Liouville equation with a spinorial Yamabe type term

TL;DR

We address blow-up issues for the super-Liouville equation with a spinorial Yamabe term on a 2D manifold. The authors develop a refined blow-up theory, including a Brezis–Merle type compactness framework, two distinct singularity types induced by the nonlinear spinor term, and energy identities for both the spinor and the scalar function, together with bubble decompositions on the plane. They prove a bubble-based energy quantization with total spinor energy decomposing into contributions from bubbles on $S^2$ and classify bubbles as spinorial Yamabe type or super-Liouville type. They further compute blow-up masses $m(p)$, establish when a unique super-Liouville bubble arises at first-type singularities, and derive a complete energy identity for the function part, showing how the total energy concentrates into bubble energies under conformal invariance and neck analysis. These results advance the understanding of coupled spinor-scalar blow-up phenomena and have implications for existence and compactness questions in conformal geometry with spinorial terms.

Abstract

In this paper, we study the super-Liouville equation with a spinorial Yamabe type term, a natural generalization of Liouville equation, super-Liouville equation and spinorial Yamabe type equation. We establish some refined qualitative properties for such a blow-up sequence. In particular, we show energy identities not only for the spinor part but also for the function part. Moreover, the local masses at a blow-up point are also computed. A new phenomenon is that there are two kinds of singularities and local masses due to the nonlinear spinorial Yamabe type term, which is different from super-Liouville equation.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof