In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

Keywords:

Hermitian harmonic map, Hermitian manifold, convex ball Hermitian harmonic map, Hermitian manifold, convex ball

MSC Classifications:

58E15, 53C07 show english descriptions Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20] 58E15 - Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
53C07 - Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20]

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