[Paper Review] Totally geodesic submanifolds of symmetric spaces, III

Bang‐Yen Chen, Pui-Fai Leung+1 more|arXiv (Cornell University)|Jul 28, 2013

Geometric Analysis and Curvature Flows14 references4 citations

TL;DR

This paper develops a representation-theoretic method to determine the stability of totally geodesic submanifolds in symmetric spaces, using the Casimir operator to analyze the Jacobi operator's eigenvalues. It establishes a stability theorem for minimal totally real submanifolds in Kählerian symmetric spaces and proves that the real Grassmannian $G^\mathbb{R}(p,q)$ is unstable in $G^\mathbb{C}(p,q)$, extending results on minimal submanifold stability via curvature and harmonic form analysis.

ABSTRACT

One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds $M_+,M_-$ introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405--425] as minimal submanifolds. The other purpose is to establish a stability theorem for minimal totally real submanifolds of Kählerian manifolds.

Motivation & Objective

• To develop a general method for determining the stability of totally geodesic submanifolds in symmetric spaces using group actions and representation theory.
• To apply this method to determine the stability of the basic submanifolds $M_{+}$ and $M_{-}$ introduced in part II of the series.
• To establish a stability theorem for minimal totally real submanifolds in Kählerian symmetric spaces.
• To analyze the second variation of volume via the Jacobi operator and relate it to curvature and harmonic forms.

Proposed method

• Use of the Casimir operator associated with a group action to simplify the spectral analysis of the Jacobi operator $L$ on the normal bundle.
• Reduction of the stability problem to representation-theoretic data by exploiting $G_N$-invariance of the normal bundle and the orthogonal complement $\mathcal{P}$ of the Lie algebra $\mathfrak{g}_N$ in $\mathfrak{g}_M$.
• Application of the second variation formula $\mathcal{V}''(\xi) = \int_N \langle L\xi, \xi \rangle \, d\text{vol}$, where $L = -\Delta^D - \hat{A} - Q$.
• Use of the identity $\int_N \left( ||\nabla u||^2 + R^N(u,u) \right) \, d\text{vol} = \int_N \left( \frac{1}{2}||d\alpha||^2 + ||\delta u||^2 \right) \, d\text{vol}$ for 1-forms dual to Killing fields.
• Comparison of eigenvalues of the Casimir operator on representations of $\mathfrak{g}_N$ and $\mathcal{P}$ to determine sign of the lowest eigenvalue of $L$.
• Application of the formula $i(f) \geq \beta_1(N)$ for the index of a Lagrangian minimal submanifold, derived from the second variation and harmonic 1-forms.

Experimental results

Research questions

• RQ1 Under what conditions is a totally geodesic submanifold $N$ in a symmetric space $M$ stable?
• RQ2 How can the stability of $M_{+}$ and $M_{-}$ be determined using representation theory and group actions?
• RQ3 What conditions ensure the stability or instability of minimal totally real submanifolds in Kählerian symmetric spaces?
• RQ4 Can the index of a minimal Lagrangian submanifold be bounded below by its first Betti number?

Key findings

• The submanifolds $M_{+}$ and $M_{-}$ are stable if the Casimir eigenvalues on $\mathcal{P}$ are non-negative, which is verified via representation-theoretic comparison.
• The minimal totally real submanifold $G^\mathbb{R}(p,q)$ is unstable in $G^\mathbb{C}(p,q)$ because $c(P) > c(\tilde{\omega}_2)$, indicating negative eigenvalues in the Jacobi operator.
• If $R^M > 0$ and $H^1(N; \mathbb{R}) \neq 0$, then $N$ is unstable, as shown by choosing a harmonic 1-form dual to a Killing field.
• If $R^M \leq 0$, then $N$ is always stable, due to the non-positivity of the curvature term in the second variation.
• The index of a compact Lagrangian minimal submanifold $f: N \to M$ satisfies $i(f) \geq \beta_1(N)$, with equality possible under additional conditions.
• The nullity and Killing nullity of a totally geodesic submanifold can be computed via representation-theoretic formulas involving $m(\lambda)$, $d_\lambda$, and $\dim \mathfrak{g}_i^\perp$.