Scalar curvature and harmonic maps to $S^1$

Scalar Curvature, Moment Maps, and the Deligne Pairing *

Let X be a compact complex manifold and L → X a positive holomorphic line bundle. Assume that Aut(X,L)/C× is discrete, where Aut(X,L) is the group of holomorphic automorphisms of the pair (X,L). Donaldson [D] has recently proved that if X admits a metric ω ∈ c1(L) of constant scalar curvature, then (X,L) is Hilbert-Mumford stable for k sufficiently large. Since Kähler-Einstein metrics have cons...

Harmonic Maps and Biharmonic Maps

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.

Positive Scalar Curvature and Surgery

Harmonic Maps to Teichmüller Space

We give sufficient conditions for the existence of equivariant harmonic maps from the universal cover of a Riemann surface B to the Teichmüller space of a genus g ≥ 2 surface Σ. The condition is in terms of the representation of the fundamental group of B to the mapping class group of Σ. The metric on Teichmüller space is chosen to be the Kähler hyperbolic metric. Examples of such representatio...

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...