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We presented a systematic treatment of a Hilbert criterion for stabilitytheory for an action of a real reductive group $G$ on a real submanifold $X$ ofa K\"ahler manifold $Z$. More precisely, we suppose the action of a compactconnected Lie group $U$ with Lie algebra $\mathfrak{u}$ extends holomorphicallyto an action of the complexified group $U^{\mathbb C}$ and that the $U$-actionon $Z$ is Hamiltonian. If $G\subset U^{\mathbb C}$ is closed and compatible,there is a corresponding gradient map $\mu_\mathfrak{p} : X\longrightarrow\mathfrak{p}$, where $\mathfrak g = \mathfrak k \oplus \mathfrak p$ is a Cartandecomposition of the Lie algebra of $G$. The concept of energy complete actionof $G$ on $X$ is introduced. For such actions, one can characterize stability,semistability and polystability of a point by a numerical criteria using a$G$-equivariant function associated with a gradient map, called maximal weightfunction. We also prove the classical Hilbert-Mumford criteria for semistabiltyand polystability conditions.  We thank the anonymous referee for carefully reading our paper and for givingsuch constructive comments which substantially helped improving the quality ofthe paper.

Stability, analytic stability for real reductive Lie groups