A class of harmonic submersions and minimal submanifolds. (English) Zbl 0978.58006
The authors introduce the notion of a pseudo-horizontally homothetic map. This denotes a smooth map \(\varphi\) from a Riemannian manifold \(M\) to a Kähler manifold \(N\) such that \(d\varphi \circ d\varphi^\ast\) commutes with the Kähler structure \(J\) and such that
\[
d\varphi(\nabla_V d\varphi^\ast(JY)) = J d\varphi(\nabla_Vd\varphi^\ast(Y))
\]
holds for all vector fields \(Y\) locally defined on \(N\) and all horizontal tangent vectors \(V\) on \(M\). This class includes holomorphic and antiholomorphic maps between Kähler manifolds.
The main result of the paper says that if \(\varphi\) is a pseudo-horizontally homothetic harmonic submersion and if \(P \subset N\) is a complex submanifold, then \(\varphi^{-1}(P)\) is a minimal submanifold of \(M\). This can for example be used to find minimal submanifolds of \(\mathbb{CP}^n\) which are not complex submanifolds.
The main result of the paper says that if \(\varphi\) is a pseudo-horizontally homothetic harmonic submersion and if \(P \subset N\) is a complex submanifold, then \(\varphi^{-1}(P)\) is a minimal submanifold of \(M\). This can for example be used to find minimal submanifolds of \(\mathbb{CP}^n\) which are not complex submanifolds.
Reviewer: Christian Bär (Hamburg)
MSC:
58E20 | Harmonic maps, etc. |
53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Keywords:
harmonic maps; minimal submanifolds; Kähler manifolds; pseudo-horizontally weakly conformal maps; pseudo-horizontally homothetic mapsReferences:
[1] | DOI: 10.1007/s002290050198 · Zbl 0938.53035 · doi:10.1007/s002290050198 |
[2] | DOI: 10.1007/BFb0096222 · doi:10.1007/BFb0096222 |
[3] | DOI: 10.1007/BF01934344 · Zbl 0757.53031 · doi:10.1007/BF01934344 |
[4] | Brnznescu V., Springer-Verlag pp 1624– (1996) |
[5] | Burns D., J. Diff. Geom. 30 pp 579– (1989) · Zbl 0678.53062 · doi:10.4310/jdg/1214443603 |
[6] | DOI: 10.1142/S0129167X97000299 · Zbl 0904.53044 · doi:10.1142/S0129167X97000299 |
[7] | DOI: 10.1112/blms/10.1.1 · Zbl 0401.58003 · doi:10.1112/blms/10.1.1 |
[8] | DOI: 10.1112/blms/20.5.385 · Zbl 0669.58009 · doi:10.1112/blms/20.5.385 |
[9] | DOI: 10.5802/aif.691 · Zbl 0339.53026 · doi:10.5802/aif.691 |
[10] | DOI: 10.1007/BF02568184 · Zbl 0826.53029 · doi:10.1007/BF02568184 |
[11] | DOI: 10.1007/BF01264019 · Zbl 0826.53028 · doi:10.1007/BF01264019 |
[12] | Gudmundsson S., Math. Scand. 73 pp 127– (1993) · Zbl 0790.58009 · doi:10.7146/math.scand.a-12460 |
[13] | Hsiang W. Y., J. Diff. Geom. 5 pp 1– (1971) · Zbl 0219.53045 · doi:10.4310/jdg/1214429775 |
[14] | DOI: 10.1142/S0129167X97000457 · Zbl 0910.58010 · doi:10.1142/S0129167X97000457 |
[15] | DOI: 10.1090/conm/049/833811 · doi:10.1090/conm/049/833811 |
[16] | DOI: 10.1142/S0129167X92000187 · Zbl 0763.53051 · doi:10.1142/S0129167X92000187 |
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