Related papers: K-stability and parabolic stability

In this note, we prove a 2-systolic inequality on compact positive scalar curvature K\"ahler surfaces admitting a nonconstant holomorphic map to a positive-genus compact Riemann surface. According to the classification of positive scalar…

An explicit expression is obtained for the sectional curvature in the plane spanned by two stationary flows, cos(k, x) and cos(l, x). It is shown that for certain values of the wave vectors k and l the curvature becomes positive for alpha >…

In odd dimensions, we prove a scalar curvature rigidity for parabolic convex polytopes in hyperbolic space enclosed by linear planes in the Poincare upper half-space model and convex with respect to the conformally related flat metric. Our…

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a…

Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kaehler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable…

Although standard planar double bubbles are stable in the sense that the second variation of the perimeter functional is non-negative for all area-preserving perturbations the question arises whether they are dynamically stable. By…

The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of…

In this new version, we correct some typos. For the readers' convenience, we also added some footnotes and more details for certain lemmas and theorems.

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…

For a polarized algebraic manifold $(X,L)$, let $T$ be an algebraic torus in the group of all holomorphic automorphisms of $X$. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by…

We analyze the classical stability of Q-tubes --- charged extended objects in $(3+1)$-dimensional complex scalar field theory. Explicit solutions were found analytically in the piecewise parabolic potential. Our choice of potential allows…

In this paper, we report a "new" continuity path which links the constant scalar curvature equation to a second order elliptic equation. This is largely an expository article where we describes various aspects of geometry and analysis…

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property,…

Let $X$ be a complex manifold and let $g$ be a polyhedral metric on it inducing its topology. We say that $g$ is a polyhedral K\"ahler (PK) metric on $X$ if it is K\"ahler outside its singular set. The local geometry of PK metrics is…

We obtain necessary conditions for the existence of special K\"ahler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we…

In this paper, we study factorable surfaces in a 3-dimensional isotropic space. We classify such surfaces with constant isotropic Gaussian (K) and mean curvature (H). We provide a non-existence result related with the surfaces satisfying…

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is…

We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow…

In this article we introduce the notion of Polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and…