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# Communications in Analysis and Geometry

## Volume 12 (2004)

### Number 3

*K* Energy and *K* Stability on Hypersurfaces

Pages: 601 – 630

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n3.a5

#### Author

#### Abstract

Suppose that *M* is a compact Fano manifold. That is, *M* is a compact Kähler manifold with positive first Chern class. One of the most important problems in Kähler geometry is the existence of Kähler metrics of constant scalar curvature. It is believed that the problem is related to certain notion of stability in the sense of Geometric Invariant Theory.

In Tian [17] and Donaldson [4], the notion of *K* stability was introduced. In the first three sections of this paper, we use the notations in [17] to derive our theorems. In the last section, we discuss the definition of [4] and some observations motivated by that paper.