

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
Bibtex
Keywords and phrases: conformal invariant, $\sgima_k$ scalar curvature, Schur Lemma
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Abstract:
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
<center class="math-display"> <img src="/fileadmin/preprint_img/2011/tex_1636a0x.png" alt="&#x222B; -- &#x222B; R- 2 R- 2 M |Ric - 3 g|dv(g) &#x2264; 9 M |Ric- 3g|dv(g), " class="math-display"></center>where R = vol(g)-1 ∫ MRdv(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
<center class="math-display"> <img src="/fileadmin/preprint_img/2011/tex_1636a1x.png" alt=" &#x222B; ^Y([g0]) := sup vol(&#x222B;g)-M-&#x03C3;2(g)dv(g), g&#x2208;C1([g0]) ( M &#x03C3;1(g)dv(g))2 " class="math-display"></center> where 1([g0]) := {g = e-2ug0 |R > 0} and prove that
([g0]) ≤ 1∕3, which implies the above inequality.