The image of the Gauss map of any oriented isoparametric hypersurface in the standard unit sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n(\mathbb{C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant principal curvatures in $S^{n+1}(1)$ is a compact monotone and cyclic embedded Lagrangian submanifold with minimal Maslov number $2n/g$. We obtain the Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces of classical type with $g = 4$. Combining with our results in [25] and [27], we completely determine the Hamiltonian stability of the Gauss images of all homogeneous isoparametric hypersurfaces.

A two-dimensional periodic Schr\"{o}dingier operator is associated with every Lagrangian torus in the complex projective plane ${\mathbb C}P^2$. Using this operator we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional $W^{-}$ introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel--Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e. preserves the value of the energy functional. In particular, the deformations generated by Novikov--Veselov equations preserve the area of minimal Lagrangian tori.

We prove that the rotation in time T of a trajectory of a K-Lipschitz vector ¯eld in Rn around a given point (stationary or non-stationary) is bounded by A + BKT with A;B absolute constants. In particular, trajectories of a Lipschitz vector ¯eld in ¯nite time cannot have an in¯nite rotation around a given point (while trajectories of a C1 vector ¯eld may have an in¯- nite rotation around a straight line in ¯nite time). The bound above extends to the mutual rotation of two trajectories (for the time intervals T and T0, respectively) of a K-Lipschitz vector ¯eld in R3: this rotation is bounded from above by the quantity CK min(T; T0) + DK2TT0.

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green function for $\sqrt{P^\prime}$.

We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Khler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and SiuYau on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the Khler cone. First, we deform the metric along the SasakiRicci flow and obtain a limit SasakiRicci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the SasakiRicci soliton to a SasakiEinstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats