Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness
Zhijie Chen, Houwang Li
TL;DR
This paper proves a sharp compactness result for positive critical points of the Moser-Trudinger functional on compact surfaces with conical singularities by performing a detailed bubbling analysis. It introduces an interpolation functional $I_{p,\beta}$ to bridge subcritical and critical regimes, and classifies blow-up into non-singular and singular bubbles, each contributing quantized energy ($4\pi$ or $4\pi(1+\alpha)$ respectively). The authors establish local gradient and energy controls using isothermal coordinates and a conformal transformation to a smooth metric, and derive precise quantization formulas for both Type I and Type II blow-up at singular points. These results provide the compactness foundation needed for proving existence of critical points in a sequel, thereby extending the Moser-Trudinger theory to conical singular surfaces and enriching the understanding of energy concentration phenomena in singular geometric settings.
Abstract
Let $(Σ, g_1)$ be a compact Riemann surface with conical singularites of angles in $(0, 2π)$, and $f: Σ\to\mathbb R$ be a positive smooth function. In this paper, by establishing a sharp quantization result, we prove the compactness of the set of positive critical points for the Moser-Trudinger functional \[F_1(u)=\int_Σ(e^{u^2}-1)f dv_{g_1}\] constrained to $u\in\mathcal E_β:=\{u\in H^1(Σ,g_1) : \|u\|_{H^1(Σ,g_1)}^2=β\}$ for any $β>0$. This result is a generalization of the compactness result for the Moser-Trudinger functional on regular compact surfaces, proved by De Marchis-Malchiodi-Martinazzi-Thizy (Inventiones Mathematicae, 2022, 230: 1165-1248). The presence of conical singularities brings many additional difficulties and we need to develop different ideas and techniques. The compactness lays the foundation for proving the existence of critical points of the Moser-Trudinger functional on conical singular surfaces in a sequel work.