相关论文: On asymptotics for the Mabuchi energy functional

We show that for any solution to the K\"ahler-Ricci flow with positive bisectional curvature on a compact K\"ahler manifold $M^n$, the bisectional curvature has a uniform positive lower bound. As a consequence, the solution converges…

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We…

Let $f:X \longrightarrow X $ be a Cohomological Hyperbolic Mapping of a complex compact connected K\"ahler manifold with $ dim_{\mathbb{C}}(X)=k \ge 1$. We want to study the dynamics of such mapping from a probabilistic point of view, that…

We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space. More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In…

For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an…

We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded domain in $\mathbf{R}^{n}$. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler…

In our previous works on deformation limits of projective and Moishezon manifolds, we introduced and made crucial use of the notion of strongly Gauduchon metrics as a reinforcement of the earlier notion of Gauduchon metrics. Using direct…

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted…

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle…

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne-Mumford stability. For every pair of integers 0<m<n, we…

In this note we prove the following result: There is a positive constant $\epsilon(n,\Lambda)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $\Lambda$, diameter…

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a…

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a…

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with…

We classify quadruples $(M,g,m,\tau)$ in which $(M,g)$ is a compact K\"ahler manifold of complex dimension $m>2$ with a nonconstant function $\tau$ on $M$ such that the conformally related metric $g/\tau^2$, defined wherever $\tau\ne 0$, is…

Let (M,g) be a compact Riemmanian manifold with non-empty boundary. Consider the second order hyperbolic initial-boundary value problem (\delta_t^2 + P(x,D))u = 0 in (0,T) x M, u(0,x) = \delta_t u(0,x) = 0 for x in M, u(t,x) = f(t,x) on…

Given a smooth closed Riemannian manifold $(M,g)$ of dimension $N \ge 3$, we derive sharp quantitative stability estimates for nonnegative functions near the solution set of the Yamabe problem on $(M,g)$. The seminal work of Struwe (1984)…

We study the spectrum and perturbative stability of Freund-Rubin compactifications on $M_p \times M_{Nq}$, where $M_{Nq}$ is itself a product of $N$ $q$-dimensional Einstein manifolds. The higher-dimensional action has a cosmological term…

We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity…

Conformally K\"{a}hler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in K\"{a}hler geometry. We introduce uniform K-stability for toric K\"{a}hler manifolds, and show that uniform K-stability is…