[Paper Review] Generalized immersions and minimal hypersurfaces in compact symmetric spaces

Ricardo A. E. Mendes, Marco Radeschi|arXiv (Cornell University)|Aug 19, 2017

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper establishes a linear lower bound on the index plus nullity of closed, immersed, minimal hypersurfaces in compact symmetric spaces, dependent on their first Betti number. It further improves this to an affine lower bound on the index alone under genericity or product ambient space conditions, using a generalized notion of isometric immersions with skew-symmetric second fundamental form.

ABSTRACT

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSes, we improve this to a lower bound on the index alone, which is affine in the first Betti number. To prove these, we introduce a generalization of isometric immersions in Euclidean space. Compact symmetric spaces admit (and in fact are characterized by) such a structure with skew-symmetric second fundamental form.

Motivation & Objective

• To establish a lower bound on the index plus nullity of closed, immersed, minimal hypersurfaces in compact symmetric spaces.
• To improve this bound to depend solely on the index under additional geometric conditions.
• To introduce a generalized framework for isometric immersions in symmetric spaces, characterized by skew-symmetric second fundamental forms.
• To characterize compact symmetric spaces via the existence of such generalized immersions.

Proposed method

• Introduce a generalization of isometric immersions in Euclidean space, adapted to compact symmetric spaces.
• Define the second fundamental form of such immersions to be skew-symmetric, linking it to the geometry of symmetric spaces.
• Use spectral theory and Jacobi operator analysis to relate the index and nullity to topological invariants like the first Betti number.
• Apply curvature and symmetry properties of compact symmetric spaces, particularly those that are products of CROSSes, to refine the index bounds.
• Utilize the structure of the ambient space to impose genericity conditions that simplify the index estimate.
• Leverage the characterization of compact symmetric spaces via the existence of such generalized immersions with skew-symmetric second fundamental forms.

Experimental results

Research questions

• RQ1 What is the minimal possible index plus nullity for a closed, immersed, minimal hypersurface in a compact symmetric space, in terms of its first Betti number?
• RQ2 Under what geometric conditions can the lower bound on index plus nullity be refined to depend only on the index?
• RQ3 How does the skew-symmetry of the second fundamental form in generalized isometric immersions relate to the symmetric space structure of the ambient manifold?
• RQ4 Can the class of compact symmetric spaces be characterized by the existence of such generalized immersions with skew-symmetric second fundamental forms?
• RQ5 What role does the product structure of the ambient space—specifically, being a product of two CROSSes—play in improving the index bound?

Key findings

• A linear lower bound on the index plus nullity of closed, immersed, minimal hypersurfaces in compact symmetric spaces is established, with the bound depending on the first Betti number.
• When the hypersurface satisfies a genericity condition or the ambient space is a product of two CROSSes, the bound improves to an affine lower bound on the index alone.
• Compact symmetric spaces are characterized by the existence of generalized isometric immersions with skew-symmetric second fundamental forms.
• The generalized immersion framework provides a new geometric tool to analyze minimal hypersurfaces in symmetric spaces.
• The index bound is affine in the first Betti number under the specified conditions, indicating a strong topological influence on spectral properties.
• The second fundamental form's skew-symmetry is a key structural feature that links the geometry of the immersion to the symmetric space's curvature and symmetry.