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

One of the main aims of this article is to give the complete classificationof critical metrics of the volume functional on a compact manifold $M$ withboundary $\partial M$ and with harmonic Weyl tensor, which improves thecorresponding classification for complete locally conformally flat case, due toMiao and Tam [18]. In particular, we prove that a critical metric with harmonicWeyl tensor on a simply connected compact manifold with boundary isometric to astandard sphere $\mathbb{S}^{n-1}$ must be isometric to a geodesic ball in asimply connected space form $\Bbb{R}^n,$ $\Bbb{H}^n$ and $\Bbb{S}^n.$ In orderto achieve our goal, firstly we shall conclude the classification of suchcritical metrics under the Bach-flat assumption and then we will prove thatboth geometric conditions are indeed equivalent.

On the volume functional of compact manifolds with boundary with harmonic Weyl tensor