Abstract
Let M^{m} be a compact convex hypersurface in R^{m+1}. In this paper, we prove that if the principal curvatures lambda_{i} of M^{m} satisfy 0<lambda_{1}leq cdots leq lambda_{m} and 3lambda_{m}<sum_{j=1}^{m-1}lambda_{j}, then there exists no nonconstant stable F-stationary map between M and a compact Riemannian manifold when (6) or (7) holds.
Highlights
Let u : (Mm, g) → (Nn, h) be a smooth map between Riemannian manifolds (Mm, g) and (Nn, h)
Kawai and Nakauchi [ ] introduced a functional related to the pullback metric u∗h as follows:
By using the second variation formula, they proved that every stable F-stationary map from Sm( ) to any Riemannian manifold is constant if u∗h F
Summary
Let u : (Mm, g) → (Nn, h) be a smooth map between Riemannian manifolds (Mm, g) and (Nn, h). The map u is stationary for if it is a critical point of (u) with respect to any compact supported variation of u, and u is stable if the second variation for the functional (u) is nonnegative. They showed the nonexistence of a nonconstant stable stationary map for , either from Sm (m ≥ ) to any manifold, or from any compact Riemannian manifold to Sn (n ≥ ). The authors in [ ] obtained the first and second variation formula for F-stationary maps.
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