This paper provides a study of some aspects of flat and curved BPS domain walls together with their Lorentz invariant vacua of four dimensional chiral N = 1 supergravity. The scalar manifold can be viewed as a one-parameter family of Kähler manifolds generated by a Kähler-Ricci flow equation. Consequently, a vacuum manifold characterized by (m,λ) where m and λ are the dimension and the index of...

We address some aspects of four-dimensional chiral N = 1 supersymmetric theories on which the scalar manifold is described by Kähler geometry and can further be viewed as Kähler–Ricci soliton generating a one-parameter family of Kähler geometries. All couplings and solutions, namely the BPS domain walls and their supersymmetric Lorentz invariant vacua turn out to be evolved with respect to the ...

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration. In [7], S. K. Donaldson proposed an ambitious program to tackle the problem of the existence and uniqueness of extremal metrics on a Kähler manifold from the perspective of the infinite dimensional space of Kähler potentials. He observed that the existenc...

In this paper, we study the uniformly strong convergence of Kähler-Ricci flow on a Fano manifold with varied initial metrics and smoothly deformed complex structures. As an application, prove uniqueness solitons in sense diffeomorphism orbits. The result generalizes Tian-Zhu’s theorem for compact manifold, it is also generalization Chen-Sun’s Kähler-Einstein metric

Geodesics on the infinite dimensional symmetric space H of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in X×A, where A ⊂ C is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces GC/G. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Ampère geodesics c...

Let X be a simply connected compact Kähler manifold with zero first Chern class, and let L be an ample line bundle over X. The pair (X,L) is called a polarized Calabi-Yau manifold. By a theorem of Mumford, the moduli space of the pair (X,L) (CY moduli) exists and is a complex variety. Locally, up to a finite cover, the moduli space is smooth (see [20, 21]). There is a natural Kähler metric, cal...

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S1 can be constructed from another hyperKähler manifold (of lower dimention) with an action of S1 which fixes one complex structure and rotates the other two and vice versa. On the other hand, the...

In [2], the weak Kähler-Ricci flow was introduced for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is allowed to be no longer Kähler. The convergence as t → 0 is of great importance to study for this topic. 1 Motivation and Set-up Kähler-Ricci flow, the complex version of Ricci flow, has been under inten...

We continue our study of heterotic compactifications on non-Kähler complex manifolds with torsion. We give further evidence of the consistency of the six-dimensional manifold presented earlier and discuss the anomaly cancellation and possible supergravity description for a generic non-Kähler complex manifold using the newly proposed superpotential. The manifolds studied in our earlier papers ha...

The famous Frankel conjecture asserts that any compact Kähler manifold with positive bisectional curvature must be biholomorphic to CP n. This conjecture was settled affirmatively in early 1980s by two groups of mathematicians independently: Siu-Yau[16] via differential geometry method and Morri [15] by algebraic method. There are many interesting papers following this celebrated work; in parti...