Curvature-decreasing maps are volume-decreasing. On joint work with G. Besson and G. Courtois.

Gallot, S.

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We study the evolution of the renormalized volume functional for even-dimensional asymptotically Poincaré-Einstein metrics (M, g) under normalized Ricci flow. In particular, we prove that [...] where S(g(t)) is the scalar curvature for the evolving metric g(t). This implies that if S +n(n − 1) ≥ 0 at t = 0, then RenV(Mn , g(t)) decreases monotonically. For odd-dimensional asymptotically Poincaré-Einstein metrics with vanishing obstruction tensor,we find that the conformal anomaly for...

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W. Roter (1987)

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Einstein manifolds, volume rigidity and Seiberg-Witten theory

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A. Nabutovsky (1995)

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Weakly-Einstein hermitian surfaces

Vestislav Apostolov, Oleg Muškarov (1999)

Annales de l'institut Fourier

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A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kähler form of an Einstein Hermitian surface is an eigenform of the curvature operator. Referring to this property as we obtain a complete classification of the compact locally homogeneous $*$ -Einstein Hermitian surfaces. We also provide large families of non-homogeneous $*$ -Einstein (but non-Einstein) Hermitian metrics on $ℂ ℙ^{2} ♯ {\overline{ℂ ℙ}}^{2}$ , $ℂ ℙ^{1} \times ℂ ℙ^{1}$ , and on the product surface $X \times Y$ of two curves $X$ and $Y$ whose genuses are greater than 1...