Related papers: An approach to Griffiths conjecture

Let $L$ be a holomorphic line bundle over a compact K\"ahler manifold $X$. Motivated by mirror symmetry, we study the deformed Hermitian-Yang-Mills equation on $L$, which is the line bundle analogue of the special Lagrangian equation in the…

Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,\theta)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,\theta)$ is polystable. A similar result for…

Yau's uniformization conjecture states: a complete noncompact K\"ahler manifold with positive holomorphic bisectional curvature is biholomorphic to $\ce^n$. The K\"ahler-Ricci flow has provided a powerful tool in understanding the…

We consider the problem of preserving positive Ricci curvature along connected sums. In this context, based on earlier work by Perelman, Burdick introduced the notion of core metrics and showed that the connected sum of manifolds with core…

We prove that, starting at an initial metric $g(0)=e^{2u_0}(dx^2+dy^2)$ on $\mathbb{R}^2$ with bounded scalar curvature and bounded $u_0$, the Ricci flow $\partial_t g(t)=-R_{g(t)}g(t)$ converges to a flat metric on $\mathbb{R}^2$.

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for…

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to…

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical K\"ahler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a…

In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits…

We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying K\"ahler manifolds. We also study the…

The periodic orbit conjecture states that, on closed manifolds, the set of lengths of the orbits of a non-vanishing vector field all whose orbits are closed admits an upper bound. This conjecture is known to be false in general due to a…

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities)…

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic…

By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kahler manifold X, the Yang-Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is…

Let M be a compact Riemannian manifold without boundary and let E be a Riemannian vector bundle over M. If $\sigma$ denotes the sphere subbundle of E, we look for embeddings of $\sigma$ into E admitting a prescribed mean curvature.

The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the $C^\infty$ topology to a critical point. The construction uses only the complex gauge group action, which leads to an…

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse…

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian…

The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential…

Let $M$ be a compact, holomorphic symplectic Kaehler manifold, and $L$ a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of $L$ is effective. This result is related to the hyperkaehler SYZ…