Related papers: Volume estimates for K\"ahler-Einstein metrics and…
In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi…
Solutions to vacuum Einstein field equations with cosmological constant, such as the de Sitter space and the anti-de Sitter space, are basic in different cosmological and theoretical developments. It is also well known that complex…
We survey selected developments in the metric geometry of the space of K\"ahler metrics, emphasizing results from the past decade, highlighting open problems along the way.
In this article we pursue the following main goals. In the first place, we establish the existence of "estimable" Hermite--Einstein metrics for stable reflexive coherent sheaves on compact normal K\"ahler spaces. If moreover the background…
We study two different natural notions of singular K\"ahler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat K\"ahler cone metrics that arise as non-collapsed limits of sequences of K\"ahler-Einstein…
We show that the singularities of the twisted K\"ahler--Einstein metric arising as the long-time solution of the K\"ahler--Ricci flow or in the collapsed limit of Ricci-flat K\"ahler metrics is intimately related to the holomorphic…
There is an obstruction to the existence of K\"ahler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to…
We propose an approach to the existence problem for locally conformally K\"ahler metrics on compact complex manifolds by introducing and studying a functional that is different according to whether the complex dimension of the manifold is…
In this talk we discuss finite-volume computations of two-body hadronic decays below the inelastic threshold (e.g. $K\to\pi\pi$ decays). In particular we show how the relation between finite-volume matrix elements and physical amplitudes,…
In this paper, I give a new construction of a K\"{a}hler-Einstein metrics on a smooth projective variety with ample canonical bundle. This result can be generalized to the construction of a singular K\"{a}hler-Einstein metric on a smooth…
We prove a local volume noncollapsing estimate for K\"ahler metrics induced from a family of complex Monge-Amp\`ere equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various…
In this paper, we establish a Liouville type rigidity result for a class of asymptotically hyperbolic non-compact Einstein metrics defined on manifolds of dimension $d\ge 5$ extending the earlier result in dimension $d=4$.
We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)K\"ahler metric. Furthermore we show that the (pseudo-)K\"ahler metrics defined on some domain in…
Using $\delta$-invariants and Newton--Okounkov bodies, we derive the optimal volume upper bound for K\"ahler manifolds with positive Ricci curvature, from which we get a new characterization of the complex projective space.
We discuss finite-volume computations of two-body hadronic decays below the inelastic threshold (e.g. $K\to\pi\pi$ decays). The relation between finite-volume matrix elements and physical amplitudes, recently derived by Lellouch and…
We construct a toric generalised K\"ahler structure on $\mathbb{C}P^2$ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised…
In this paper, we discuss the self-shrinking systems in higher codimensional spaces. We mainly obtain several Bernstein type results and a sharp growth estimate.
We derive some elliptic differential inequalities from the Weitzenb\"ock formulas for the traceless Ricci tensor of a K\"ahler manifold with constant scalar curvature and the Bochner tensor of a K\"ahler-Einstein manifold respectively.…
In this paper, we derive apriori estimates for constant scalar curvature K\"ahler metrics on a compact K\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\"ahler potential. We also…
We calculate Chow quotients of some families of symmetric \(T\)-varieties. In complexity two we obtain new examples of K\"ahler-Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional…