[Paper Review] Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups

Shinji Ohno, Takashi Sakai+1 more|arXiv (Cornell University)|Dec 4, 2016

Geometric Analysis and Curvature Flows3 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for orbits of commutative Hermann actions and products of symmetric subgroups in compact Lie groups and symmetric spaces to be biharmonic, using symmetric triads with multiplicities. It classifies all proper biharmonic submanifolds in irreducible compact symmetric spaces (singular orbits, cohomogeneity two) and biharmonic hypersurfaces in compact simple Lie groups (regular orbits, cohomogeneity one), providing explicit lists of such submanifolds across multiple classical and exceptional Lie groups.

ABSTRACT

We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducible symmetric spaces of compact type which are singular orbits of commutative Hermann actions of cohomogeneity two. Also, in compact simple Lie groups, we determine all the biharmonic hypersurfaces which are regular orbits of actions of the direct product of two symmetric subgroups which are associated to commutative Hermann actions of cohomogeneity one.

Motivation & Objective

• To characterize biharmonic orbits in compact symmetric spaces and Lie groups using symmetric triads with multiplicities.
• To determine all proper biharmonic submanifolds that are singular orbits of commutative Hermann actions of cohomogeneity two in irreducible symmetric spaces of compact type.
• To classify all biharmonic hypersurfaces that are regular orbits of actions of the product of two symmetric subgroups associated with cohomogeneity one commutative Hermann actions in compact simple Lie groups.
• To extend the understanding of biharmonic submanifolds beyond Euclidean space, particularly in non-positively curved and compact symmetric target spaces.
• To resolve cases where biharmonic submanifolds are not minimal, providing counterexamples to generalized Chen’s conjecture in positive curvature settings.

Proposed method

• The authors use the theory of symmetric triads with multiplicities to describe the second fundamental forms of orbits in compact symmetric spaces and Lie groups.
• They derive a criterion for biharmonicity based on the vanishing of the bitension field, expressed via the second fundamental form and curvature tensor.
• The method relies on the classification of commutative compact symmetric triads (G, K₁, K₂) with dim 𝔞 = 1 and G simple, using root system data and multiplicities m(α), m(2α), n(α), n(2α).
• The biharmonic condition is reduced to a polynomial equation in the multiplicities: (m₁ + 6m₂ + 6n)² − 64(m₁ + m₂)n = 0.
• The authors analyze this equation across all known compact symmetric triads of rank one, identifying cases where solutions exist.
• They apply this criterion to classify all biharmonic hypersurfaces in compact simple Lie groups and singular orbits in symmetric spaces of compact type.

Experimental results

Research questions

• RQ1 Which orbits of commutative Hermann actions of cohomogeneity two in irreducible compact symmetric spaces are proper biharmonic submanifolds?
• RQ2 For which compact simple Lie groups and actions of (K₂ × K₁) on G via symmetric subgroups is the regular orbit biharmonic but not minimal?
• RQ3 Does the generalized B.Y. Chen’s conjecture hold in compact symmetric spaces of non-negative curvature, and if not, what are the counterexamples?
• RQ4 What is the complete list of proper biharmonic hypersurfaces in compact simple Lie groups arising as regular orbits of (K₂ × K₁)-actions associated with cohomogeneity one Hermann actions?
• RQ5 Under what conditions on multiplicities (m₁, m₂, n) does the biharmonic condition (m₁ + 6m₂ + 6n)² − 64(m₁ + m₂)n = 0 admit non-trivial solutions?

Key findings

• All proper biharmonic submanifolds in irreducible symmetric spaces of compact type that are singular orbits of cohomogeneity two commutative Hermann actions are classified, with examples in SO(1+b+c), SU(4), Sp(2), SO(2+2q), and E₆.
• In compact simple Lie groups, all biharmonic hypersurfaces that are regular orbits of (K₂ × K₁)-actions associated with cohomogeneity one Hermann actions are classified, including cases in SO(1+b+c), SU(1+b+c), Sp(1+b+c), and F₄.
• For (G, K₁, K₂) = (SO(6), U(3), SO(3)×SO(3)) and (SU(1+q), SO(1+q), S(U(1)×U(q))) with 52 ≥ q > 1, all biharmonic orbits are harmonic, showing the conjecture fails only for specific parameter ranges.
• The paper identifies 14 distinct types of symmetric triads where proper biharmonic hypersurfaces exist, including (E₆, SO(10)·U(1), F₄) and (F₄, Sp(3)·Sp(1), Spin(9)).
• For (Sp(1+q), U(1+q), Sp(1)×Sp(q)), proper biharmonic hypersurfaces exist only when q = 2 or q > 45, indicating a sharp threshold.
• The condition (m₁ + 6m₂ + 6n)² − 64(m₁ + m₂)n = 0 has no solution for compact symmetric triads with dim 𝔞 = 1 and G simple, ruling out biharmonic submanifolds in certain symmetric settings.