The linear stability of non-Kähler Calabi-Yau metrics

Non-Kähler Calabi-Yau theory is a newly developed subject and it arises naturally in mathematical physics and generalized geometry. The relevant geometries are pluriclosed metrics which are critical points of the generalized Einstein–Hilbert action which is an extension of Perelman's $F$-functional. In this work, we study the critical points of the generalized Einstein-Hilbert action and discuss the stability of critical points which are defined as pluriclosed steady solitons. We proved that all compact Bismut–Hermitian–Einstein metrics are linearly stable which is non-Kähler analogue of the stability results of Ricci solitons from Tian, Zhu [27] and Hall, Murphy [10], Koiso [13]. In addition, all compact Bismut-flat pluriclosed metrics with $(2n-1)$-positive Ricci curvature are strictly linearly stable when the complex structure is fixed.