arXiv research

Study pseudo-Riemannian Sasaki metrics on solvable Lie groups.

The paper constructs Einstein Sasaki metrics on solvable Lie groups.

New Einstein metric found on non-standard solvmanifold.

In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curva…

We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply …

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.

Study compact symplectic solvmanifolds' hard Lefschetz property.

This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show tha…

We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explic…

Exploring Sasaki metrics in joined manifolds.

Characterizes hypercomplex Lie groups and their solvmanifolds.

We obtain new supersymmetric flux vacua of type II supergravities on four-dimensional Minkowski times six-dimensional solvmanifolds. The orientifold O4, O5, O6, O7, or O8-planes and D-branes are localized. All vacua are in addition not T-dual to a vacuum on the torus. The corresponding solvmanifolds are proven to be Ca…

The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.

New Einstein solvmanifolds created from non-flat Ricci solitons.

Constructs mirror pairs for solvmanifolds using Lie groups.

We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). Applying our main result we will list all possible maps on infra-so…

The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $\varphi$ such that the induced metric $g_{\varphi}$ is Einstein, unless $g_{\varphi}$ is flat.…

Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.

In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.

We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).

Compact Kähler solvmanifolds are classified up to biholomorphism. A proof of a conjecture Benson and Gordon, that completely solvable compact Kähler solvmanifolds are tori is deduced from this. The main ingredient in the proof is a restriction theorem for polycyclic Kähler groups proved by Nori and the author.

We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds). It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds. J. Heber has showed that under certain simple algebraic condition called standard (i.e. the orthogonal complement of the deriv…

Surveying mean curvature flow on solvmanifolds, focusing on translating solutions.

Negative curvature proven in Sasaki manifold space completion.

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.

New Einstein solvmanifolds constructed without using nilsolitons.

The study classifies flat solvmanifolds and finds $G_2$-structures on them.

We consider the stability of Sasaki-extremal metrics under deformations of the complex structure on the Reeb foliation. Given such a deformation preserving the action of a compact subgroup of the automorphism group of a Sasaki-extremal structure, a sufficient condition is given involving the nondegeneracy of the relati…

A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…

We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some rem…

New curvature for weighted Sasaki sphere found.

New approach linking CR Yamabe invariant to Sasaki structures.

Study 1-Lefschetz contact solvmanifolds, proving their characterization.

The paper explores existence and non-existence of constant scalar curvature and extremal Sasaki metrics.

The paper computes non-trivial triple Massey products on specific non-Kähler solvmanifolds.

The study explores indefinite nilsolitons and Einstein solvmanifolds, revealing new geometric properties.

Let $G/H$ be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold $Γ\big\backslash G/H$ is by definition a quotient of $G/H$ by a discrete uniform subgroup $Γ\leq G$. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $…

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisma…

We show the equivalence of several definitions of compact infra-solvmanifolds that appear in various math literatures.

Study of Sasaki groups vs Kähler groups, showing distinct behaviors.

In this paper we give a diameter bound for Sasaki manifolds with positive transverse Ricci curvature. As an application, we obtain the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure.

The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.

The article classifies 6D flat solvmanifolds by analyzing conjugacy classes of matrices.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is eas…

In this paper, we introduce a class of Sasaki manifolds with a reductive $G$-group action, called $G$-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessar…

We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bound…