Equivariant Morse Index

• Equivariant Morse index is a refined invariant that measures instability of critical manifolds using group-invariant perturbations.
• It arises in variational problems for minimal hypersurfaces, employing techniques from min–max theory and equivariant versions of the classical Morse index.
• Key results include sharp index bounds and applications to Morse inequalities, with implications for stability and compactness in symmetric geometric settings.

An equivariant Morse index generalizes the notion of Morse index for critical submanifolds or hypersurfaces subject to symmetry, measuring instability only with respect to equivariant (group-invariant) perturbations. This concept is central in the variational theory of minimal hypersurfaces under group actions and in equivariant Morse theory, reflecting the interplay between topology, symmetry, and variational instability. The equivariant Morse index appears as a key invariant in min–max theory, equivariant Morse (co)homology, index theorems, and equivariant Morse inequalities.

1. Definition and Characterization of the Equivariant Morse Index

Let $(M^{n+1},g)$ be a closed Riemannian manifold with a compact Lie group $G$ acting isometrically. For a closed, embedded, $G$–invariant minimal hypersurface $\Sigma\subset M$, the Jacobi operator (second variation operator) $L_\Sigma$ acts on the space of compactly supported normal vector fields $\mathcal X^\perp(\Sigma)$. The classical Morse index, $\mathrm{Index}(\Sigma)$, counts negative eigenvalues of $L_\Sigma$ on $\mathcal X^\perp(\Sigma)$.

Equivariant Morse Index: Restricting attention to the $G$–invariant normal vector fields $\mathcal X^{\perp,G}(\Sigma)$, the $G$–Morse index is defined as the number of negative eigenvalues of $L_\Sigma$ on this subspace: $$\mathrm{Index}_G(\Sigma) = \#\{\text{negative eigenvalues of } L_\Sigma \text{ on } \mathcal X^{\perp,G}(\Sigma)\}$$ A minimal hypersurface is $G$–stable if $\mathrm{Index}_G(\Sigma)=0$. The index quantifies the dimension of the maximal $G$–invariant subspace where the second variation is negative-definite. This definition aligns precisely with Definition 2.7 in (Wang, 2022).

For Morse functions on $G$–manifolds with finite $G$, the equivariant Morse index at a critical point $p$ with stabilizer $G_p$ is given as the dimension of the negative Hessian eigenspace $V_p^-$ as a $G_p$–representation, thus integrating representation-theoretic data into the definition (Bao et al., 2024, Lu, 2012). For Morse–Bott critical submanifolds $B_i$, $\mathrm{ind}_G(B_i)$ is the sum of the dimensions (or representations) corresponding to negative eigenvalues of the Hessian restricted to the normal bundle.

In the $S^1$–equivariant context, the S$^1$–equivariant Morse index is the dimension of the negative eigenspace of the Hessian, restricted to the normal space of the orbit of the group action (Zadeh et al., 2014).

2. Equivariant Min–Max Theory and Index Bounds

Equivariant min–max theory constructs $G$–invariant minimal hypersurfaces using $k$–parameter sweepouts within the space of $G$–invariant cycles. The min–max value is

$$\omega^G_k(M) := \inf_\Phi \sup_{x\in\mathrm{dom}\,\Phi} \mathrm{Area}(\Phi(x)),$$

and the limiting varifold is shown to be a smooth, embedded, minimal $G$–hypersurface under regularity assumptions: $3\leq \min_{p\in M}\mathrm{codim} G\cdot p\leq 7$.

Equivariant Index Bound Theorem (Wang, 2022): If a min–max hypersurface $\Sigma$ arises from a $k$–dimensional $G$–homotopy class, one obtains

$\mathrm{Index}_G(\Sigma)\leq k,$

with equality achieved for well-chosen sweepouts, showing sharpness of the bound in many cases.

Key technical tools include:

• The equivariant bumpy–metrics theorem: under generic $G$–invariant metrics, no minimal $G$–hypersurface admits nontrivial $G$–invariant Jacobi fields beyond those from group action itself.
• Compactness theorems showing smooth graphical convergence of min–max $G$–hypersurfaces, with area and index bounds.
• Deformation theorems to bypass stationary varifolds in the critical set with index $\geq k+1$ by flows generated by negative-definite $G$–invariant Jacobi fields.
• Existence of $G$–invariant Jacobi fields when the limiting object has multiplicity $>1$, ensuring index lower bounds and enforcing multiplicity-one convergence for generic metric.

For free-boundary minimal surfaces with maximal symmetry group $G$, analogous min–max constructions yield lower and upper equivariant index bounds. For example, for minimal disc stackings in $B^3$, $\mathrm{ind}_G\Sigma_{N,m}\geq\lfloor N/2\rfloor$, showing that high equivariant Morse index can force the necessity of multi-parameter sweepouts in equivariant min–max realizations (Carlotto et al., 2023).

3. Comparison with Classical Morse Index

The $G$–equivariant Morse index always satisfies the inequality

$$\mathrm{Index}_G(\Sigma)\leq \mathrm{Index}(\Sigma),$$

as $G$–invariant variations form a proper subspace of all normal variations. Therefore, $G$–stability is stronger than classical stability: a hypersurface can have positive classical Morse index (i.e., classical instability), but be $G$–stable if no $G$–invariant destabilizing directions exist.

Conversely, if $\mathrm{Index}(\Sigma)\geq k$, then $$\mathrm{Index}_G(\Sigma)\geq \min\{k,\dim\mathcal X^{\perp,G}(\Sigma)\}$$. This relation positions the equivariant index as coarser, often sharper in detecting instability relevant to group-symmetric perturbations (Wang, 2022).

In Morse–Bott theory, the equivariant Morse index at a $G$–invariant critical submanifold or orbit is also a sum of the dimensions of negative eigenspaces under $G$–representation, highlighting its representation-theoretic nature (Lu, 2012).

4. Illustrative Examples and Sharpness

• Reflection in the Sphere: For $G=\mathbb{Z}_2$ acting on $S^n$ by reflection, the equator $S^{n-1}$ is minimal with classical Morse index $1$ but has \emph{no} nontrivial $\mathbb{Z}_2$–invariant normal field; so $\mathrm{Index}_G=0$.
• Rotation on $S^3$: For $G=S^1$ acting by coordinate rotation, the Clifford torus has $\mathrm{Index}_G$ strictly less than its classical Morse index, as only certain Fourier modes are $S^1$–invariant.
• Disc Stackings: Minimal disc stackings with $N$ layers in $B^3$ and symmetry group $G$ have equivariant Morse index at least $\lfloor N/2\rfloor$. This demonstrates that 1-parameter equivariant min–max constructions only capture cases $N=2,3$; higher $N$ require more parameters (Carlotto et al., 2023).

In these settings, the equivariant index bound in min–max theory is often sharp; equality can occur for certain $k$–parameter families of sweepouts (Wang, 2022).

5. Equivariant Morse Index in Equivariant Morse Theory

For $G$–invariant Morse functions $f$ on a closed manifold with $G$ action (finite or compact Lie), the equivariant Morse index at a critical point $p$ can be defined as $\dim_\mathbb{R} V_p^-$, where $V_p^-$ is the negative eigenspace of the Hessian (viewed as a $G_p$–representation), with $G_p$ the isotropy group of $p$. This definition integrates representation-theoretic data, leading to richer invariants.

Stable critical points (all negative directions invariant under the stabilizer action) produce descending manifolds fixed by $G_p$, yielding clean $G$–CW decompositions. The resulting equivariant Morse complex has basis labeled by $G$–orbits of critical points, and the boundary operator counts $G$–orbits of flow lines, with representation-theoretic weighting (Bao et al., 2024).

This structure leads to a spectral sequence converging to $H^G_*(M)$, with the $E^1$–page freely generated by orbits of critical points of given equivariant index.

6. Applications in Index Theory and Equivariant Morse Inequalities

Equivariant versions of Morse inequalities, both in the finite group and compact Lie group settings, encode lower bounds for the equivariant Betti numbers in terms of equivariant Morse data.

For a $G$–invariant Morse–Bott function $f$ on a closed manifold $M$ with a compact Lie group $G$ action, the equivariant Morse numbers $m_j^G(f)$ and Betti numbers $b_j^G(M)$ (as virtual $G$–representations) satisfy

$$\sum_{j=0}^k (-1)^{k-j} m_j^G(f) \succeq \sum_{j=0}^k (-1)^{k-j} b_j^G(M)$$

for each $k$, with equality in the full alternating sum, where $\succeq$ denotes preorder in the representation ring $R(G)$ (Lu, 2012).

For $S^1$–actions, the $S^1$–equivariant Morse index and corresponding Morse inequalities are proven using Witten deformation and heat kernel methods. The index of the $m$–th Fourier component on a complex manifold with boundary is controlled by explicit equivariant characteristic forms, giving precise index growth rates and Morse inequalities reflecting both Morse index and group symmetry (Zadeh et al., 2014, Hsiao et al., 2017).

7. Construction of $G$–Invariant Jacobi Fields and Role in Compactness

A central technical point in equivariant min–max limits is the construction of nontrivial $G$–invariant Jacobi fields when convergence occurs with multiplicity $m>1$. In such settings:

• Away from singular orbits, graphical convergence yields $m$ ordered graphs, and the difference between the top and bottom sheets satisfies a linear elliptic PDE with coefficients controlled away from singularities.
• A normalized limit yields a nontrivial $G$–invariant solution of $L_\Sigma h=0$.
• Existence of such a Jacobi field enforces $\mathrm{Index}_G(\Sigma)\geq 1$, excluding certain high-multiplicity and high-index phenomena under generic metrics, and underpins the index upper bound in min–max theory (Wang, 2022).

This mechanism further guarantees that under generic metrics, min–max sequences converge to multiplicity-one minimal hypersurfaces, which are nondegenerate in the $G$–equivariant sense.

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Key References:

• “Equivariant Morse index of min-max $G$-invariant minimal hypersurfaces” (Wang, 2022)
• “Morse homology and equivariance” (Bao et al., 2024)
• “Analytic approach to $S^1$-equivariant Morse inequalities” (Zadeh et al., 2014)
• “Equivariant Morse inequalities and applications” (Lu, 2012)
• “Disc stackings and their Morse index” (Carlotto et al., 2023)
• “$S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary” (Hsiao et al., 2017)