Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness
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arXiv
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2025
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| _version_ | 1866912381480730624 |
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| author | Chen, Zhijie Li, Houwang |
| author_facet | Chen, Zhijie Li, Houwang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness Chen, Zhijie Li, Houwang Analysis of PDEs 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. |
| title | Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.12602 |