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- W2952390030 abstract "By improving the analysis developed in the study of $s_k$-Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if $(M^3, g)$ is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, then [int_M |Ric-frac{bar R} 3 g|^2 dv (g)le 9int_M |Ric-frac{R} 3 g|^2dv(g), ] where $bar R=vol (g)^{-1} int_M R dv(g)$ is the average of the scalar curvature $R$ of $g$. Equality holds if and only if $(M^3,g)$ is a space form. We in fact study the following new conformal invariant [ds widetilde Y([g_0]):=sup_{gin {cal C}_1([g_0])}frac {ds vol(g)int_M s_2(g) dv(g)} {ds (int_M s_1(g) dv(g))^2}, ] where ${cal C}_1([g_0]):={g=e^{-2u}g_0,|, R>0}$ and prove that $widetilde Y([g_0])le 1/3$, which implies the above inequality." @default.
- W2952390030 created "2019-06-27" @default.
- W2952390030 creator A5054085447 @default.
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- W2952390030 date "2011-03-20" @default.
- W2952390030 modified "2023-10-18" @default.
- W2952390030 title "A new conformal invariant on 3-dimensional manifolds" @default.
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- W2952390030 doi "https://doi.org/10.48550/arxiv.1103.3838" @default.
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