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 σ_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 (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="&amp;#x222B; -- &amp;#x222B; R- 2 R- 2 M |Ric - 3 g|dv(g) &amp;#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=" &amp;#x222B; ^Y([g0]) := sup vol(&amp;#x222B;g)-M-&amp;#x03C3;2(g)dv(g), g&amp;#x2208;C1([g0]) ( M &amp;#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.