Related papers: A generalized Frankel conjecture via the Yang-Mill…
We derive a generalization of the flat space Yang's and Newman's equations for self-dual Yang-Mills fields to (locally) conformally Kahler Riemannian 4-manifolds. The results also apply to Einstein metrics (whose full curvature is not…
In this paper, we establish characterizations of variational $s$-convexity and tilt stability for prox-regular functions in the absence of subdifferential continuity via quadratic bundles, a kind of primal-dual generalized second-order…
We introduced a new flow to the LYZ equation on a compact K\"ahler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins-Jacob-Yau's condition on the subsolution, the longtime…
After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds.…
We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.
Transversally K\"ahler foliations are a generalisation of K\"ahler manifolds, appearing naturally in the complex non-K\"ahler setting. We give a self-contained proof of how the classical methods used in the proof of the Aubin-Yau theorem…
Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain…
We study ancient mean curvature flows in $\mathbb{R}^{n+1}$ whose tangent flow at $-\infty$ is a shrinking cylinder $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. We prove that the cylindrical profile function…
In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which…
A notion of asymptotically conical K\"ahler orbifold is introduced, and, following previous existence results in the setting of asymptotically conical manifolds, it is shown that a certain complex Monge-Amp\'ere equation admits a rapidly…
In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on K\"ahler manifolds of nonnegative orthogonal bisectional curvature.
We study the K\"ahler-Ricci flow on compact K\"ahler manifolds whose canonical bundle is big. We show that the normalized K\"ahler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges…
We study Minkowski supersymmetric flux vacua of type II string theory. Based on the work by M. Grana, R. Minasian, M. Petrini and A. Tomasiello, we briefly explain how to reformulate things in terms of Generalized Complex Geometry, which…
We consider the global aspects of the 6-dimensional $\mathcal{N}=(1, 0)$ theory arising from the coupling of the vector multiplet to the tensor multiplet. We show that the Yang-Mills field and its dual, when both are abelianized, combine to…
In this note we consider submersions from compact manifolds, homotopy equivalent to the Eschenburg or Bazaikin spaces of positive curvature. We show that if the submersion is nontrivial, the dimension of the base is greater than the…
In this paper, we generalize Huber's finite point conformal compactification theorem to a higher dimensional manifold, which is conformally compact with $L^\frac{n}{2}$ integrable Ricci curvatures.
In this paper, we introduce a new parabolic equation on K\"ahler manifolds. The static point of this flow is related to the existence of a lower bound of the Mabuchi energy. In this paper, we prove the flow always exists for all times for…
The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…
Recently, B.Chow and R.S.Hamilton introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a $S^{2}$ bundle…
A recent theorem of Diverio--Trapani and Wu--Yau asserts that a compact K\"ahler manifold with a K\"ahler metric of quasi-negative holomorphic sectional curvature is projective and canonically polarized. This confirms a long-standing…