Related papers: Volume estimates for K\"ahler-Einstein metrics: th…

In this paper we prove a local curvature estimate for the $\kappa$-LYZ flow over K\"ahler manifolds introduced in [FGP1] and [LYZ]. In particular, we generalize the long time existence of the flow.

Let $K$ be a convex body in $\R^d$, let $j\in\{1, ..., d-1\}$, and let $\varrho$ be a positive and continuous probability density function with respect to the $(d-1)$-dimensional Hausdorff measure on the boundary $\partial K$ of $K$. Denote…

We prove the local classification of K\"ahler metrics with constant holomorphic sectional curvature by exploiting the geometry of the bundle of 1-jets of holomorphic functions.

We study the complete Kahler-Einstein metric in tube domains. We obtain estimates of this metric and its holomorphic bisectional curvatures near the weakly pseudoconvex boundary points.

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge--Amp\`ere type for the stream function, when…

We prove the convergence of K\"ahler-Ricci flow with some small initial curvature conditions. As applications, we discuss the convergence of K\"ahler-Ricci flow when the complex structure varies on a K\"ahler-Einstein manifold.

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general…

We survey recent results on the existence of K\"ahler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties.…

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different…

We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of…

We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not…

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…

We show an Uhlenbeck type estimate for closed simply connected manifolds which provides the existence of certain exact sequences in K-area homology. This leads to the behavior of the K-area homology under surgery. Moreover, we give an index…

We prove the convergence of geodesic distance during the quantization of the space of K\"ahler potentials. As applications, this provides alternative proofs of certain inequalities about the K-energy functional in the projective case.

We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics…

The purpose of this paper is to derive volume and other geometric information for three-dimensional complete manifolds with positive scalar curvature. In the case that the Ricci curvature is nonnegative, it is shown that the volume of the…

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several…

We are interested in the impact of entropies on the geometry of a hypersurface of a Riemannian manifold. In fact, we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold, provided some geometric…

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of…

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…