Preprint 11/2011

A new conformal invariant on 3-dimensional manifolds

Yuxin Ge and Guofang Wang

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Submission date: 30. Mar. 2011
Pages: 25
published in: Advances in mathematics, 249 (2013), p. 131-160 
DOI number (of the published article): 10.1016/j.aim.2013.09.009
Keywords and phrases: conformal invariant, $\sgima_k$ scalar curvature, Schur Lemma
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By improving the analysis developed in the study of σk-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 (M3,g) is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, then

∫        --          ∫          R- 2                 R-  2  M |Ric - 3 g|dv(g) ≤ 9 M |Ric- 3g|dv(g),

where R = vol(g)-1MRdv(g) is the average of the scalar curvature R of g. Equality holds if and only if (M3,g) is a space form. We in fact study the following new conformal invariant

                     ∫ ^Y([g0]) :=  sup   vol(∫g)-M-σ2(g)dv(g),         g∈C1([g0]) ( M σ1(g)dv(g))2

where C1([g0]) := {g = e-2ug0|R > 0} and prove that Y^([g0]) 13, which implies the above inequality.

18.10.2019, 02:14