In this thesis, we prove a classification of pluriharmonic isometric immersions of a K¨ahler manifold in a semi-Euclidean space, which establishes a generalization of the Calabi-Lawson theory concerning minimal surfaces in Euclidean spaces. Gromoll, and the author that the geometry of pluriharmonic isometric immersions of K¨ahler manifolds in Euclidean spaces has many properties in common with that of minimal surfaces. Gromoll [10] proved that for a pluriharmonic isometric immersion f : M →RP of a simply connected K¨ahler manifold M in the Euclidean space RP, there exists a holomorphic isometric immersion Φ :M →CP such that
Namely, he has constructed a parametrization of the modulus space of complete isometric pluriharmonic immersions of a simply connected K¨ahler manifold in a semi-Euclidean space, which is described in terms of some complex matrix defined by a holomorphic of full isometric. immersion of the K¨ahler manifold in a complex semi-Euclidean space. Abe [2] proved that a pluriharmonic isometric immersion of a complete K¨ahler manifold in a real codimension Euclidean space is a cylinder (Proposition 4.1.11). This asserts that the study of pluriharmonic isometric immersions of a full K¨ahler man in a real codimension Euclidean space can be reduced to that of minimal surfaces in R3.
Rodriguez classified isometric pluriharmonic immersions with real co-dimension two of complete Kühler manifolds in Euclidean spaces in terms of the index of relative nullity, that is, the dimension of the kernel of the shape operator (definition 4.1.1 and theorem 4.1.14) . . Let M be a complete K¨ahler manifold with real size 2 m and f: M → RNN+P an isometric pluriharmonic immersion of M in RNN+P.
Semi-Euclidean spaces
Fundamental theory of isometric immersions
Let M be a simply connected d-dimensional Riemannian manifold and π : E → M a vector group over M of rank N + P − d with one metric. An isometric immersion kef :M →RN+PN is called m-cylindrical if there exists an N (d−m)-dimensional Riemannian manifold and an isometric immersion f :N →RN+PN −m such that. The following separation theorem for isometric immersions of Riemannian product manifolds in Euclidean spaces is due to J.
In a similar way, we also obtain the following cylinder theorem for isometric immersions of the product of a Riemannian manifold and a Euclidean space in semi-Euclidean spaces. Let f : N × Rm → RNN+P be an isometric immersion of the product of a Riemann manifold and Rm in a semi-Euclidean space. An isometric immersion f is called nicely curved if the dimension of the oscillating space Osckf(x) is constant for all x ∈ M and for each.
Let f : M → RN+PN be an isometric immersion of a d-dimensional Riemannian manifold in a semi-Euclidean space and L a nondegenerate submanifold of Nor f of orderq. Note that the subset L⊥ consisting of the orthogonal complementL⊥(x) iL(x) in Nor f(x) is also parallel with respect to∇⊥.
Pluriharmonic immersions
A pluriharmonic immersion is often called a (1,1)-geodesic immersion, whose name derives from condition (ii) as above. If one of the following (D) or (U) holds, then f is holomorphic with respect to some orthogonal complex structure of R2m+2. It is well known that a simply connected minimal surface in R3 has the so-called associated family, which is represented as the real part of a holomorphic immersion in C3.
The family fθ is called the adjoint family of f, in particular fπ/2 is called the conjugate immersion of f. In fact, by the definition of pluriharmonic immersions, we have g(AθξX, Y) =α(Jθ/2X, Jθ/2Y), ξRN+P. We now construct isometric immersionsfθ using Proposition 2.2.2. In fact, since the second basic form α of f satisfies the Gaussian equation,. which means that αθ satisfies the Gaussian equation.
As a consequence of this result, we have the following proposition, which will play a key role in Chapter 3.
Holomorphic immersions
Let (U;z1,. . , zm) be a local complex coordinate of M, and Φ : M → CP an isometric holomorphic immersion nicely curved on U.
Classification theorem
If f : M → RNN+P is a full isometric pluriharmonic immersion, then there exists an (N+P)×(n+p)-complex matrix S such that. We proceed to prove that (S3) is equivalent to (P4). Step 3B) For these complex numbers ai, bi, cj, dj and the matrix. Step 3C combined with Step 3D now implies that (S3) and (P4) are equivalent, completing the proof of the lemma.
With these preparations, we obtain a parameterization of the modulus space of complete isometric pluriharmonic immersions [18, 19]. Before closing this section, we now consider the module space without assuming the completeness of the submersions. Let M(M;RNN+P) denote the set of O(N, P) congruence classes of pluriharmonic isometric immersions of a K¨ahler manifold M in RN+PN.
Examples
To summarize, we have a K¨ahler manifold biholomorphic to C2 and an isometric minimal immersion f : C2 → R6 that have the following properties. 2 Re Φ is an isometric minimal immersion of a simple K¨ahler manifold M in RN+P(= RN+P0 ), where Φ : M → CN+P is an isometric holomorphic immersion such that. In summary, to obtain (locally defined) pluriharmonic immersions in RNN+P, we need to construct only holomorphic immersions in CN+P that satisfy the condition.
As an example, we will construct pluriharmonic immersions of subgroups of C2 in R51, which are defined as cone immersions.
Cylinder theorem
Let ⊥ be the distribution on G given by the orthogonal complement⊥(x) of (x) with respect to the Riemannian metric of M. For an isometric immersion of a complete Riemannian manifold in a semi-Euclidean space, the relative nullity foliation is complete. To prove our cylinder theorem, we first define the splitting tensor field for an isometric immersion, more precisely, for its relative nullity distribution.
Let its relative null distribution be on G, where the relative null index is constant ν0. Let f : M → RN+PN be an isometric immersion of a Riemannian manifold into a semi-Euclidean space with a splitting tensor C. When the relative null foliation is complete, we obtain the following property for splitting tensors.
We now prove that the splitting tensor of an isometric pluriharmonic immersion is complex linear. We are now going to prove the following cylinder theorem for isometric pluriharmonic immersions, under appropriate assumptions about the index of relative nullity and the completeness of K¨ahler manifolds. Let M be a complete K¨ahler manifold of real dimension 2m, and f :M →RN+PN an isometric pluriharmonic immersion.
Let G be an open set on which the index of relative nullity ν of f is 2m−2. If f : M → R2m+1N is an isometric immersion of real co-dimension one, then the index of relative nullity ν of f is not less than 2m−2. Before we proceed to the case of codimension two, the following definition is in order.
Let f : M → RN+PN be an isometric immersion of a K¨ahler manifold M with real dimension 2m in RNN+P. Rodriguez [14] proves the following theorem, which classifies minimal isometric immersions of complete K¨ahler manifolds in Euclidean spaces of real codimension two. Let M be a complete K¨ahler manifold with real dimension 2m≥4, and def :M →R2m+2a minimal isometric immersion of real codimension two with relative nullity index ν.
Bernstein property
Roughly speaking, the following statement means that an isometric pluriharmonic immersion is completely geodesic if its tangent vectors are separated from the orthogonal complement of a timelike vector. 1] Abe, K., A characterization of fully geodesic submanifolds in SN and CPN by an inequality, Tˆohoku Math. 6] Calabi, E., Quelques application de l'analyse complex aux Surfaces d'aire minima, Topics in complex manifolds (by Rossi, H.), Univ.
18] Furuhata, H., Construction and classification of isometric minimal immersions of K¨ahler manifolds in Euclidean spaces, Bull. 19] Furuhata, H., Moduli space of isometric pluriharmonic immersions of K¨ahler manifolds in indefinite Euclidean spaces, Pacific J.
