In this talk, we give a complete classification of Yamabe soliton, Ricci-Bourguignon soliton on real hypersurfaces in complex hyperbolic quadric ${Q^m}^*$ and complex hyperbolic space $CH^m$. Next as applications, we also give a complete classification of gradient Yamabe soliton and gradient Ricci-Bourguignon soliton on real hypersurfaces in complex hyperbolic quadric ${Q^m}^*$ and complex hyperbolic space $CH^m$. Related to this topics, finally we want to mention soliton problems and Fisher-Marsden conjecture in other Hermitian symmetric spaces like complex two-plane Grassmannians, complex hyperbolic two-plane Grassmannians, complex quadric or etc.

In this talk. first, from the histrorical view point we introduce some works on isometric Reeb flow on real hypersurfaces in complex projective space, complex hyperbolic space, complex two-plane Grassmannians, and complex quadrics. Next we want to give a recent classification of isometric Reeb flow on real hypersurfaces in generalized complex k-plane Grassamnnians.

In this talk, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ or in complex hyperbolic two-plane Grassmannians $G_2^{*}(\mathbb{C}^{m+2})$. Next by using the isometric Reeb flow we give a complete classification for hypersurfaces $M$ in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, complex hyperbolic two-plane Grassmannians $G_2^{*}(\mathbb{C}^{m+2})$ and a complex quadric $\mathbb{Q}^m$.