The first Chern class and conformal area for a twistor holomorphic immersion

Abstract

We obtain an inequality involving the first Chern class of the normal bundle and the conformal area for a twistor holomorphic surface. Using this inequality, we can improve an inequality obtained by T. Friedrich for the Euler class of the normal bundle of a twistor holomorphic surface in the fourdimensional space form. Moreover, as a corollary, we see that the area of a superminimal surface in the unit sphere is an integer multiple of 2π, which is essentially proved by E. Calabi.

Twistor holomorphic affine surfaces and projective invariants

“…The volume of superminimal surface in an even dimensional unit sphere is an integer multiple of 2π, which is essentially proved in [2]. See also [5]. Using Theorem 4.3, we can obtain this result.…”

“…Then it is interesting to study affine immersions with holomorphic twistor lifts, which are invariant under projective transformations of the ambient manifolds. Using the decomposition of connections (see [5]), we obtain several projective invariants for affine immersions with twistor lifts. Consequently, an affine immersion with holomorphic twistor lift can be characterized by the vanishing of some of these invariants.…”

“…In this section, we summarize the fundamental results for the decomposition of connection on a complex vector bundle (see [5]) and prove several lemmas which we use in the later sections. Let M be an almost complex manifold with an almost complex structure J and E a vector bundle over M with I ∈ Γ(End(E)) satisfying I 2 = −id and D ∈ C(E).…”