On the Liouville theorem for harmonic maps

Choi, Hyeong In

doi:10.1090/S0002-9939-1982-0647905-3

On the Liouville theorem for harmonic maps
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by Hyeong In Choi PDF

Proc. Amer. Math. Soc. 85 (1982), 91-94 Request permission

Abstract:

Suppose $M$ and $N$ are complete Riemannian manifolds; $M$ with Ricci curvature bounded below by $- A$, $A \geqslant 0$, $N$ with sectional curvature bounded above by a positive constant $K$. Let $u:M \to N$ be a harmonic map such that $u(M) \subset {B_R}({y_0})$. If ${B_R}({y_0})$ lies inside the cut locus of ${y_0}$ and $R < \pi /2\sqrt K$, then the energy density $e(u)$ of $u$ is bounded by a constant depending only on $A$, $K$ and $R$. If $A = 0$, then $u$ is a constant map.

References

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