The $σ_k$-Yamabe problem revisited
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910231377739776 |
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| author | Ge, Yuxin Wang, Guofang Wei, Wei |
| author_facet | Ge, Yuxin Wang, Guofang Wei, Wei |
| contents | In this paper we revisit the $σ_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $σ_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $σ_2$-Yamabe constant
$$ Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M σ_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} $$
is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $σ_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has
$$ \inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M σ_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, σ_2(g)>0} \frac{\int_M σ_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} . $$
We also show that these conclusions can fail if the condition $R_g>0$ is removed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05414 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $σ_k$-Yamabe problem revisited Ge, Yuxin Wang, Guofang Wei, Wei Differential Geometry Analysis of PDEs In this paper we revisit the $σ_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $σ_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $σ_2$-Yamabe constant $$ Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M σ_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} $$ is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $σ_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has $$ \inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M σ_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, σ_2(g)>0} \frac{\int_M σ_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} . $$ We also show that these conclusions can fail if the condition $R_g>0$ is removed. |
| title | The $σ_k$-Yamabe problem revisited |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2605.05414 |