Conformal capacities and extremal metrics

Conformal Capacities and Extremal Metrics

For any non-compact Riemannian manifold M of dimension n ≥ 2 we previously defined a function λM : M×M → R+ = R+∪{+∞] only dependent on the conformal structure of M , and proved that for a class of manifolds containing all the proper subdomains of R, λ 1 n M was a distance on M [F1, F2]. The case of a domain G of R has been the object of several investigations leading to estimations of λG[V1, ....

Extremal Metrics for Conformal Curvatures on S

We define two conformal structures on S which give rise to a different view of the affine curvature flow and a new curvature flow, the “Qcurvature flow”. The steady state of these flows are studied. More specifically, we prove four sharp inequalities, which state the existences of the corresponding extremal metrics.

Extremal eigenvalues of the Laplacian in a conformal class of metrics : the ” conformal spectrum ”

Let M be a compact connected manifold of dimension n endowed with a conformal class C of Riemannian metrics of volume one. For any integer k ≥ 0, we consider the conformal invariant λk(C) defined as the supremum of the k-th eigenvalue λk(g) of the Laplace-Beltrami operator ∆g, where g runs over C. First, we give a sharp universal lower bound for λk(C) extending to all k a result obtained by Fri...

Extremal Length and Conformal Capacityo

Extremal Metrics and Geometric Stability

This paper grew out of my lectures at Nankai Institute as well as a few other conferences in the last few years. The purpose of this paper is to describe some of my works on extremal Kähler metrics in the last fifteen years in a more unified way. In [Ti4], [Ti2], the author developed a method of relating certain stability of underlying manifolds to Kähler-Einstein metrics. A necessary and new c...