English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove the main conjecture from [M. R. Douglas, B. Shiffman and S.Zelditch, Critical points and supersymmetric vacua, II: Asymptotics andextremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerningthe metric dependence and asymptotic minimization of the expected number\mathcal{N}^{crit}_{N,h} of critical points of random holomorphic sections ofthe Nth tensor power of a positive line bundle. The first non-topological termin the asymptotic expansion of \mathcal{N}^{crit}_{N,h} is the the Calabifunctional multiplied by the constant \be_2(m) which depends only on thedimension of the manifold. We prove that \be_2(m) is strictly positive in alldimensions, showing that the expansion is non-topological for all m, and thatthe Calabi extremal metric, when it exists, asymptotically minimizes\mathcal{N}^{crit}_{N,h}.

Metric Dependence and Asymptotic Minimization of the Expected Number of Critical Points of Random Holomorphic Sections