In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is asymptotically uniformly distributed. We pay special attention to the case of locally harmonic manifolds.
In this paper we prove a matrix Li-Yau-Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate can be seen as an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds, or the complex analogue of the estimate for Riemannian manifolds obtained previously by the author.
In this article we study point configurations minimizing the discrete energy on a compact Riemannian manifold, where the energy kernel is taken to be the Green's function for the Laplacian. We show that every point in a minimizing configuration lies inside an open set called harmonic ball where no other point can enter, and that the minimum distance between any two distinct points has the optimal asymptotic order. We compute explicit bounds for the minimum distance in the case of Compact Rank One Symmetric Spaces.
We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture.
Uniform $L^1$ and lower bounds are obtained for the Green's function on compact Kähler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on Kähler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an $L^q$ norm for the volume form for some $q>1$. The proof relies on auxiliary Monge-Ampère equations, and is fundamentally non…
From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions
Max-Planck Institut for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）；ScaDS.AI, Dresden/Leipzig（萨克森州数据科学与决策支持人工智能中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be described by a multigraph associated to it. Specializing into the case of dimension 4, we classify the fixed point data. Moreover, we prove that there exist oriented $S^1$-manifolds with these fixed point data. Finally, we show that a certain multigraph behaves like a manifold.
Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space, unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green's function when in addition the Ricci curvature of the manifold is assumed to be asymptotically n…
On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can be viewed as a generalization of the recent result by Munteanu-Wang \cite{MunteanuWang2021} in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.
We consider Toeplitz operators associated with the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigenfunctions of a self-adjoint Toeplitz operator under assumption that its principal symbol has a non-degenerate minimum with discrete wells. As an application, we prove upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit.
We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafaï, Hardy and Maïda on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green …
