Prescribed scalar curvature equation on 𝑆ⁿ in the presence of reflection or rotation symmetry

Leung, Man Chun; Zhou, Feng

doi:10.1090/S0002-9939-2014-11993-9

Prescribed scalar curvature equation on $S^n$ in the presence of reflection or rotation symmetry
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by Man Chun Leung and Feng Zhou PDF

Proc. Amer. Math. Soc. 142 (2014), 1607-1619 Request permission

Abstract:

Using the flow equation for the conformal scalar curvature equation, we present existence theorems in cases where the prescribed function $\mathcal {K}$ exhibits reflection or rotation symmetry (with fixed point set denoted by $\mathcal {F}$). We also demonstrate that the “one bubble” condition, namely, \[ \displaystyle {(\max _{S^n} \mathcal {K})^{\tau } \ < \ 2 \cdot (\max _{ \mathcal {F} } \mathcal {K})^{\tau }},\] cannot be totally taken away. Here ${\tau ={1\over {2}} (n - 2).}$

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