Stability of twistor lifts for surfaces in four-dimensional manifolds as harmonic sections

Surfaces in self-dual Einstein manifolds and their twistor lifts

“…for all X ∈ T M, where u 1 , u 2 is an orthonormal frame of M. The following theorem holds (see [11]).…”

“…In this note, we report some results for such surfaces in self-dual Einstein manifolds which are proved in [10], [11] and [12].…”

Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections

“…If certain conditions for the curvature tensors are satisfied and the twistor lift is a harmonic section, then the mean curvature vector field is a holomorphic section of the normal bundle with respect to the Koszul-Malgrange holomorphic structure (see [3], [8] and [9]). In general, holomorphic sections are not parallel.…”

“…To prove the main result, we use some fundamental results obtained in the author's papers [8] and [9]. For…”

“…Surfaces with holomorphic twistor lifts are called twistor holomorphic surfaces (see [7]). On the other hand, the author studies surfaces whose twistor lifts are harmonic sections in [8] and [9]. If the ambient spaces are self-dual Einstein, the twistor lifts of twistor holomorphic surfaces are harmonic sections.…”