Stationary maps into the sphere omitting a totally geodesic subsphere of codimension two

Chen, Min

doi:10.1090/proc/15248

Stationary maps into the sphere omitting a totally geodesic subsphere of codimension two
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Proc. Amer. Math. Soc. 149 (2021), 889-896 Request permission

Abstract:

In this paper, we attempt to weaken the assumption of minimizing maps in Theorems 2, 3, 4 and Corollary 5 in [J. Differential Geom. 21 (1985), pp. 151–162]. We will prove these theorems still hold for stationary maps. We obtain the regularity for stationary maps (Theorems 1.1, 1.2). Since we can construct nonconstant stationary maps from $\mathbb {R}^k$ to $S^n$ which are bounded away from a totally geodesic subsphere of codimension two (Example 1.4), we need a stability assumption to establish a Liouville theorem for stationary maps. More generally, we deduce the Liouville theorem for stationary p-harmonic maps (Theorem 1.7).

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