Related papers: Volume estimates for K\"ahler-Einstein metrics: th…
In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we…
This paper is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into K\"ahler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
In this paper, we establish the sharp $k$-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions…
The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.
Let $M$ be a complete K\"ahler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume…
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the…
We prove estimates for the sectional curvature of hyperkaehler quotients and give applications to moduli spaces of solutions to Nahm's equations and Hitchin's equations.
The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski…
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of…
In recent years, there are many progress made in K\"ahler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal K\"ahler metrics, as well as obstructions to the existence of such metrics in…
The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation…
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness…
Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $C^{2,\alpha}$-topology. In dimension…
In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for…
This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of K\"ahler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations…
We introduce new probabilistic and variational constructions of (twisted) K\"ahler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional…
We develop a local version of Huisken-Stampacchia iteration, using it to obtain local versions of a host of important sharp curvature pinching estimates for mean curvature flow. The local estimates we obtain do not depend on the quality of…
Let us consider a projective manifold and $\Omega$ a volume form. We define the gradient flow associated to the problem of $\Omega$-balanced metrics in the quantum formalism, the \Omega$-balacing flow. At the limit of the quantization, we…
We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced…