Some properties of the Schouten tensor and applications to conformal geometry
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Abstract
The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript k Superscript plus"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Gamma _k^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that this eigenvalue condition for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to n slash 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \geq n/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:msub> <mml:mi> σ </mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\sigma _k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.
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