Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness

TL;DR

This paper proves the compactness of positive critical points of a Moser-Trudinger functional on conical singular surfaces, extending known results from smooth surfaces and addressing challenges posed by singularities.

Contribution

It establishes a sharp quantization and compactness result for the Moser-Trudinger functional on surfaces with conical singularities, a significant generalization of previous smooth surface results.

Findings

Proved compactness of critical points set for the functional

Established sharp quantization in the presence of singularities

Extended classical results to conical singular surfaces

Abstract

Let $(\Sigma, g_1)$ be a compact Riemann surface with conical singularites of angles in $(0, 2\pi)$, and $f: \Sigma\to\mathbb R$ 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 \[F_1(u)=\int_{\Sigma}(e^{u^2}-1)f dv_{g_1}\] constrained to $u\in\mathcal E_\beta:=\{u\in H^1(\Sigma,g_1) : \|u\|_{H^1(\Sigma,g_1)}^2=\beta\}$ for any $\beta>0$. 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.