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Diameter estimates for K\"ahler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $$L^\infty$$ estimates for the Monge-Amp\`ere equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time existence of the K\"ahler-Ricci flow and finite-time solutions when the K\"ahler class is big, as well as for special vibrations of Calabi-Yau manifolds. 

Abstract The stable reduction theorem says that a family of curves of genus$$g\ge 2$$ $g\ge 2$over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new this result for curves defined over$${\mathbb {C}}$$ $C$, using the Kähler–Einstein metrics on the fibers to obtain the limiting stable curves at the punctures. 

We continue our work on the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler–Ricci flow with conical singularities along a divisor with simple normal crossings. 

Under optimal tournament design, equilibrium effort is invariant to the shape of the mean-zero additive stochastic component, often referred to as a “shock” or “noise”. We report data from laboratory experiments providing the first test of this prediction. Consistent with theory, we find that average effort does not significantly differ between a negatively skewed and uniform shock distribution. In addition, we test a second theoretical prediction that, in winner tournaments, when the shock distribution is asymmetric as in our design, one should exert minimum effort whenever one’s competitors are exerting above equilibrium effort. With a symmetric shock distribution as in our design, efforts should generally remain substantial, even when one’s competitors are exerting effort above equilibrium value. Our data reveal that subjects actively engage in the tournament even when faced with aggressive competitors under both shock distributions. 

Abstract Let 𝑋 be a Kähler manifold with semiample canonical bundle $K_X$K_{X}.It is proved in [W. Jian, Y. Shi and J. Song, A remark on constant scalar curvature Kähler metrics on minimal models,Proc. Amer. Math. Soc.147(2019), 8, 3507–3513] that, for any Kähler class 𝛾, there exists $\delta >0$\delta>0such that, for all $t\in (0,\delta )$t\in(0,\delta), there exists a unique cscK metric $g_t$g_{t}in $K_X+t\gamma$K_{X}+t\gamma.In this paper, we prove that ${\left\{(X,g_t)\right\}}_{t\in (0,\delta )}$\{(X,g_{t})\}_{t\in(0,\delta)}have uniformly bounded Kähler potentials, volume forms and diameters.As a consequence, these metric spaces are pre-compact in the Gromov–Hausdorff sense.