On $G_2$-structures, special metrics and related flows

Abstract

We review results about G2-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.

Geometric Partial Differential Equations from Unified String Theories

“…We note in this context the wealth of results for supersymmetric solutions of supergravity theories in dimensions 5 and 6 [41,42,44,49,6]. In a different vein, interesting related flows of balanced metrics or G 2 structures have been proposed in [5,7,54,60,30]. Yet another line of investigation is that of static solutions with symmetry, pioneered in [88] for Einstein's equations in 4 dimensional, and more recently in [1] for 5-dimensional minimal supergravity.…”

Laplacian coflow for warped G2-structures

“…In order to obtain solutions of the Laplacian flow of a warped closed G 2 -structure, combining the expressions (9) and Proposition 3.1, we can set the system of equations that must be satisfied: where απ 2 ptq ´βσ 2 ptq " 0.…”

“…They also reinterpret the flow as a set of evolution equations on M 6 involving the differential forms defining the SUp3q-structure and the warping function f . More details about the Laplacian flow of a closed G 2 -structure can be found in the reviews [9,16] and the references therein. Another interesting result concerning this flow was due to Xu and Ye in [23], where they proved long time existence and uniqueness of solution for this flow starting near a torsion free G 2 -structure.…”

Parabolic complex Monge-Ampère equations on compact Hermitian manifolds

“…Motivated by unified string theories and mirror symmetry, there has recently been interest in extending the theory of such equations beyond the Kähler setting to more general classes of Hermitian manifolds [27,24,17,10,12,13,14,19,20,26,2,11].…”