Autodual Einstein versus Kahler-Einstein

Metriques autoduales sur la boule

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishe…

LCK metrics on Oeljeklaus-Toma manifolds versus Kronecker's theorem

A locally conformally Kähler (LCK) manifold is a manifold which is covered by a Kähler manifold, with the deck transform group acting by homotheties. We show that the search for LCK metrics on Oeljeklaus-Toma manifolds leads to a (yet another) variation on Kronecker's theorem on units. In turn, this implies that on Oeljeklaus-Toma manifold associated to number fields with $2t$ complex embeddings and $s$ real embeddings with $s<t$ there is no LCK metric.

Mabuchi Kähler 孤立子与极值 Kähler 度量及更进一步的比较

Mabuchi Kähler solitons versus extremal Kähler metrics and beyond

本文利用Han-Li的v-孤立子Yau-Tian-Donaldson型对应，证明若Fano簇具有标量曲率严格小于2(n+1)的极值Kähler度量，则存在Mabuchi孤立子，从而以极值度量的存在性刻画Mabuchi孤立子，并推广至v-孤立子。