Some new obstructions to minimal and Lagrangian isometric immersions. (English) Zbl 1026.53009

Chen, Bang-Yen

Jap. J. Math., New Ser. 26, No. 1, 105-127 (2000).

The author presents a refinement of his \(\delta\)-invariant for isometric immersions. Let \(M\) be an \(n\)-dimensional Riemannian manifold with sectional curvature \(K\). For every subspace \(L\subset T_pM\) with \(\dim L\geq 2\) he defines \(\tau(L):= \sum_{i<j} K(e_i\wedge e_j)\), where \(e_1,\dots,e_r\) denotes an orthonormal basis of \(L\). In particular, \(\tau(T_pM)\) is the scalar curvature of \(M\) at \(p\). Furthermore he defines \[ \delta(n_1, \dots,n_k) (p):= \tau(T_pM)- \inf\left\{ \sum^k_{i=1} \tau(L_i)\mid L_i\subset T_pM,\;\dim L_i= n_i,\;L_i\perp L_j\right\} \] (for each family \((n_1,\dots,n_k)\) for which this definition makes sense) and numbers \(b(n_1,\dots,n_k)\in\mathbb{Z}\) and \(c(n_1,\dots, n_k)\in \mathbb{Q}\) (only depending on \((n_1,\dots,n_k))\). The main result is the following theorem: If \(M\) is a submanifold of a real space form \(R^m( \varepsilon)\) of constant curvature \(\varepsilon\) and \(H\) denotes the mean curvature of \(M\), then the inequality \[ \delta(n_1,\dots,n_k)\leq c(n_1,\dots, n_k)\cdot H^2 +b(n_1,\dots,n_k) \cdot\varepsilon \] is always true, and equality holds if and only if the shape operator of \(M\) is of a special kind. The same result is true for totally real submanifolds in complex space forms of constant, holomorphic sectional curvature \(4\varepsilon\). The curvature invariants \(\delta(2,\dots,2)\) are of particular interest. Several applications are described.

Reviewer: H.Reckziegel (Köln)

MSC:

53B25	Local submanifolds
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

curvature invariants; submanifolds of real space forms; submanifolds of complex space forms; sectional curvature; shape operator