In these notes of the mini-course given by the author at the XI-th Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2009 after the introduction of the toric Kähler geometry, we present Calabi's family of ${\rm U}(n)$-invariant extremal Kähler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.

We determine surfaces of genus zero in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections. As corollary, we prove that a surface of genus zero in four-dimensional Euclidean space is twistor holomorphic if its twistor lift is harmonic section.

In 1691, James (Jacob) Bernoulli proposed a problem called elastica problem: What shape of elastica, an ideal thin elastic rod on a plane, is allowed? Daniel Bernoulli discovered its energy functional, Euler-Bernoulli energy function, and the minimal principle of the elastica. Using it, Euler essentially solved the problem in 1744 by developing the variational method, elliptic integral theory and so on. This article starts with a review of its mathematical meaning and historical background. After that we present one of its extensions, statistical mechanics of elastica as a model of the DNA and the large polymers. We will call it a quantized elastica, and show that it is connected with the modified Korteweg-de Vries hierarchy, loop space, submanifold Dirac operators, moduli spaces of the real hyperelliptic curves and so on. By reviewing the other extensions of the elastica problem, we will see that elastica is in the center of mathematics even now.

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.