Related papers: Stability of line bundle mean curvature flow

We prove that the existence of a $Z$-positive and $Z$-critical Hermitian metric on a rank 2 holomorphic vector bundle over a compact K\"ahler surface implies that the bundle is $Z$-stable. As particular cases, we obtain stability results…

Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano…

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power…

We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold $\hat{X}$ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the…

We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is…

We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical K\"ahler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl K\"ahler manifold, which is asymptotic to a $\mu$-stable holomorphic vector…

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then…

In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a K\"{a}hler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable K\"{a}hler current with singularities in higher…

In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $\lambda$ of the maximum and the minimum of the holomorphic sectional curvatures…

We study the Fubini-Study currents and equilibrium metrics of continuous Hermitian metrics on sequences of holomorphic line bundles over a fixed compact K\"ahler manifold. We show that the difference between the Fubini-Study currents and…

A recent paper (arxiv.org:1810.00025) studied properties of a compactification of the moduli space of irreducible Hermitian-Yang-Mills connections on a hermitian bundle over a projective algebraic manifold. In this follow-up note, we show…

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-H\"older-type decomposition of (special)…

In this paper, we give some convergence results of Lagrangian mean curvature flow under some stability conditions in a general K\"ahler-Einstein manifold. In particular, we prove that the flow will converge if the initial data is some small…

Over a compact K\"ahler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a K\"ahler metric with conic singularities along a divisor. We deduce several existence results of constant scalar…

Consider $E$ a holomorphic vector bundle over a projective manifold $X$ polarized by an ample line bundle $L$. Fix $k$ large enough, the holomorphic sections $H^0(E\otimes L^k)$ provide embeddings of $X$ in a Grassmanian space. We define…

Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor…

We prove that the parabolic flow of conformally balanced metrics introduced by Phong, Picard and Zhang in "A flow of conformally balanced metrics with K\"ahler fixed points", is stable around Calabi-Yau metrics. The result shows that the…

We study equations on a principal bundle over a compact complex manifold coupling connections on the bundle with K\"ahler structures in the base. These equations generalize the conditions of constant scalar curvature for a K\"ahler metric…

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $ E $ be a holomorphic vector bundle over a compact K\"ahler manifold $(M,\omega_g) $. Suppose that there exists a smooth Hermitian metric $ h_0 $ on $E$ such…

Given a proper, open, holomorphic map of K\"ahler manifolds, whose general fibers are Calabi-Yau manifolds, the volume forms for the Ricci-flat metrics induce a hermitian metric on the relative canonical bundle over the regular locus of the…