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The equality case in Wu-Yau inequalities

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 Added by Stefano Trapani
 Publication date 2017
  fields
and research's language is English




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In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kahler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.



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139 - Haojie Chen , Xiaolan Nie 2021
We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.
64 - Alexey Balitskiy 2015
In this note we apply the billiard technique to deduce some results on Viterbos conjectured inequality between volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbos conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbos conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.
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We consider the natural generalization of the parabolic Monge-Amp`ere equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperkahler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Amp`ere equation introduced by Alesker and Verbitsky. The result gives an alternative proof of a theorem of Alesker.
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