Related papers: On the transverse scalar curvature of a compact Sa…
We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of K\"ahler metrics with constant scalar curvature (csck) on compact K\"ahler…
In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo…
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K\"ahler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays,…
We discuss a natural extension of the K\"ahler reduction of Fujiki and Donaldson, which realises the scalar curvature of K\"ahler metrics as a moment map, to a hyperk\"ahler reduction. Our approach is based on an explicit construction of…
We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal K\"ahler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the…
We discuss the existence and non-existence of constant scalar curvature, as well as extremal, Sasaki metrics. We prove that the natural Sasaki-Boothby-Wang manifold over the admissible projective bundles over local products of non-negative…
In this expository article we review the problem of finding Einstein metrics on compact K\"ahler manifolds and Sasaki manifolds. In the former half of this article we see that, in the K\"ahler case, the problem fits better with the notion…
We introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus $\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth (weight)…
Donaldson conjectured \cite{Dona96} that the space of K\"ahler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and…
We propose a new approach to the existence of constant transversal scalar curvature Sasaki structures drawing on ideas and tools from the CR Yamabe problem, establishing a link between the CR Yamabe invariant, the existence of Sasaki…
We introduce a notion of scalar curvature of a twisted generalized Kahler manifold in terms of pure spinors formalism. A moment map framework with a modified action of generalized Hamiltonians on an arbitrary compact generalized Kahler…
We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible K\"ahler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including…
We prove that constant scalar curvature K\"ahler metric "adjacent" to a fixed K\"ahler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T…
In this paper, by generalizing the concept of balanced metrics, we shall show that Donaldson's asymptotic approximation of balanced metrics for constant scalar curvature cases can be generalized to extremal Kaehler cases.
We describe a general procedure for constructing new Sasaki metrics of constant scalar curvature from old ones. Explicitly, we begin with a regular Sasaki metric of constant scalar curvature on a 2n+1-dimensional compact manifold M and…
Extending a result of He to the non-integrable case of K-contact manifolds, it is shown that transverse Hermitian scalar curvature may be interpreted as a moment map for the strict contactomorphism group. As a consequence, we may generalize…
We show that the existence of constant scalar curvature K\"ahler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the…
In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when…
In this paper we consider the Boothby-Wang construction over twist 1 stage 3 Bott orbifolds given in terms of the log pair $(S_{\bf n},\Delta_{\bf m})$. We give explicit constant scalar curvature (CSC) Sasaki metrics either directly from…
We present an infinite-dimensional hyperk\"ahler reduction that extends the classical moment map picture of Fujiki and Donaldson for the scalar curvature of K\"ahler metrics. We base our approach on an explicit construction of hyperk\"ahler…