Related papers: Extremal Kaehler-Einstein metric for two-dimension…

We conjecture that any scalar-flat K\"ahler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite…

It is shown that any smooth strictly convex global solution of $$\det(\frac{\partial^{2}u}{\partial \xi_{i}\partial \xi_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial \xi_{i}} - d_0\right\},$$ where $d_0$, $d_1$,...,$d_n$…

In this work, optimal rigidity results for eigenvalues on K\"ahler manifolds with positive Ricci lower bound are established. More precisely, for those K\"ahler manifolds whose first eigenvalue agrees with the Ricci lower bound, we show…

Two results regarding K\"ahler supermanifolds with potential $K=A+C\theta\bar\theta$ are shown. First, if the supermanifold is K\"ahler-Einstein, then its base (the supermanifold of one lower fermionic dimension and with K\"ahler potential…

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding…

The requirement that a (non-Einstein) K\"ahler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local…

We consider the massive scalar-tensor theory in the Jordan frame $F(\Phi) =K^{2}\Phi^2$ and $U(\Phi) =(1/2)m^{2}\Phi^2$, where $F(\Phi)$ corresponds to a constant Brans-Dicke parameter $\omega_{BD}=1/4K^2$. The constraint of the Solar…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…

We give a complete list of non-isometric bidimensional rotation invariant K\"ahler-Einstein submanifolds of a finite dimensional complex projective space endowed with the Fubini-Study metric. This solves in the aforementioned case a…

The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically…

For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in…

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete K\"{a}hler-Einstein metric if and only if its canonical bundle is positive. We consider the…

As part of a programme to classify quasi-Einstein metrics $(M,g,X)$ on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field $X$ is divergence-free but not identically zero. This…

The simplex was conjectured to be the extremal convex body for the two following "problems of asymmetry":\\ P1) What is the minimal possible value of the quantity $\max_{K'} |K'|/|K|$? Here, $K'$ ranges over all symmetric convex bodies…

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta\to 0$ of K\"ahler-Einstein metrics $\omega_\beta$ with a cone singularity of angle $2\pi…

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It…

Let {(M,g_i)} be a sequence of smooth compact oriented Einstein 4-manifolds of fixed Einstein constant $\lambda > 0$ that Gromov-Hausdorff converges to a 4-dimensional Einstein orbifold X. Suppose, moreover, that the limit metric is…

In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of…

We study singular K\"ahler-Einstein metrics that are obtained as non-collapsed limits of polarized K\"ahler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the…