Related papers: On asymptotics for the Mabuchi energy functional

Let $X$ be a compact Gauduchon manifold, and let $E$ and $V_0$ be holomorphic vector bundles over $X$. Suppose that $E$ is stable when considering all subsheaves preserved by a Higgs field $\theta\in H^0($End$(E)\otimes V_0)$. Then a…

In their proof of the positive energy theorem, Schoen and Yau showed that every asymptotically flat spacelike hypersurface M of a Lorentzian manifold which is flat along M can be isometrically imbedded with its given second fundamental form…

Multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles. Given a compact Riemannian manifold (X,g), its real homology H_*(X,R) is naturally endowed with the stable norm. Briefly, if h\in…

Let $X$ be a compact K\"ahler manifold and $\alpha$ a K\"ahler class on $X$. We prove that if $(X,\alpha)$ is uniformly K-stable for models, then there is a unique cscK metric in $\alpha$. This was first proved in the algebraic case by Chi…

We study the weighted constant scalar curvature, a modified scalar curvature introduced by Lahdili depending on weight functions $(v, w)$, on certain non-compact semisimple toric fibrations, a generalization of the Calabi Ansatz defined by…

The real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to…

The rigidity of the Riemannian positive mass theorem for asymptotically hyperbolic manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the hyperbolic space. This leads to study the…

Let $E$ be a vector bundle over a smooth curve $C$, and $V$ a generating space of sections of $E$. We characterise Mumford linear stability of the associated projective model of $\mathbb{P} E^\vee$ in $\mathbb{P} V^\vee$ in terms of…

In this paper, we motivate and define $\Phi$-energy density, $\Phi$-energy, $\Phi$-harmonic maps and stable $\Phi$-harmonic maps. Whereas harmonic maps or $p$-harmonic maps can be viewed as critical points of the integral of $\sigma_1$ of a…

Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's $\nu$-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the…

We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are all nearly…

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of…

In this article we study the stability problem for positive quaternion-K\"ahler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of…

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing energy sub-critical nonlinearity, associated with a family of solitary waves which…

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form…

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the…

Hermitian positive definite, totally positive, and nonsingular M-matrices enjoy many common properties, in particular: (A) positivity of all principal minors, (B) weak sign symmetry, (C) eigenvalue monotonicity, (D) positive stability. The…

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group $K$ in a Hamiltonian fashion. We prove that the squared moment map $||\mu||^2$ is constant if and only if $K$ is semisimple and the manifold is…

Consider a Hamiltonian action by biholomorphisms of a compact Lie group $K$ on a Kaehler manifold $X$, with moment map $\mu:X\to\klie^*$. We characterize which orbits of the complexified action of $G=K^{\CC}$ in $X$ intersect $\mu^{-1}(0)$…

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb].…