Lefschetz seminar, Clark University — 2021.11.12

Willie Wai Yeung Wong

wongwwy@math.msu.edu

Michigan State University

*Slides available* : https://slides-n-notes.qnlw.info

- Joint work with:
- Audrey Rosevear,
*Amherst College* - Samuel Sottile,
*Michigan State University*

- Audrey Rosevear,
- Support by NSF through the SURIEM (2020) summer REU site at MSU.
- Paper to appear in
*La Matematica*; arXiv:2101.09266.

Audrey will present a poster on this at JMM2022; do stop by and say hi!

- Set-up
- Physics
- Geometry; prior results
- Our results
- Application; what's next?

- $M(n)$ — space of $n\times n$ real matrices
- $S(n)$ — symmetric elements of $M(n)$
- $SL(n)$ — determinant 1 elements of $M(n)$
- $SO(n)$ — elements of $SL(n)$ satisfying $A^T = A^{-1}$
- $\mathring{S}_+(n)$ — positive definite elements of $S(n) \cap SL(n)$
- $\mathfrak{sl}(n)$ — trace 0 elements of $M(n)$
- $\mathfrak{so}(n)$ — anti-symmetric elements of $M(n)$
- $\langle A, B\rangle := \mathrm{tr}(AB^T)$ — Hilbert-Schmidt inner product on $M(n)$

$M(n)$ with $\langle A,B\rangle$ $\qquad\cong\qquad$ $\mathbb{R}^{n\times n}$ with standard metric.

$SL(n)$ is a co-dimension 1;

induced Riemannian metric — the HS metric.

__Not__ bi-invariant!

(Non-compact, but semi-simple, so Killing form is pseudo-Riemannian.)

What is the geometry of this Riemannian manifold?

- Geodesic completeness? — ✓
- Existence / stability of bounded geodesics?
- Asymptotic behavior and stability of unbounded geodesics?

Why emphasis on geodesics?

- Initial/reference body: \(\quad \Omega\Subset\mathbb{R}^n\)

Center of mass condition: \(\displaystyle \quad \int_\Omega y ~\mathrm{d}y = 0\) - Configuration at time \(t\): $\quad x(t) + A(t)\cdot\Omega; \qquad x(t)\in \mathbb{R}^n, \quad A\in SO(n)$
- Total kinetic energy: \[ \int_{\Omega} |x'(t) + A'(t)y|^2 ~\mathrm{d}y = \mathrm{vol}(\Omega) |x'(t)|^2 + \mathrm{tr}\left( A'(t) I_\Omega A'(t)^T \right)\]
- Moment of inertia: \(\quad \displaystyle I_\Omega = \int_\Omega y y^T ~ \mathrm{d}y\)

For write-up with more details: click here.

\[ S = \int \left[ \mathrm{vol}(\Omega) |x'(t)|^2 + \mathrm{tr}\left( A'(t) I_\Omega A'(t)^T \right) \right] ~dt\]

The integrand defines a positive definite quadratic form on the tangent space of $\mathbb{R}^n\times SO(n)$

Solutions are geodesics w.r.t. this Riemannian metric.*Note that the metric given by $I_\Omega$ is not the bi-invariant metric on $SO(n)$.*

Motion on $\mathbb{R}^n$ decouples from motion on $SO(n)$.

- Allow the body to also deform linearly but preserving volume.
- Swap $SO(n)$ with $SL(n)$.
- Everything else the same: $I_\Omega$ induces a Riemannian metric on $SL(n)$ which determines the trajectories.

Why do we only care when $I_\Omega = \mathrm{Id}_n$?

- Allow the body to arbitrarily deform, but preserving infinitesimal volume.
- Replace $SO(n)$ by the group(oid) of volume preserving diffeomorphisms.
- For fluid flow inside a fixed domain $\Omega$: Euler-Arnol'd equations.
- Free boundary problem: requires boundary conditions.
- At any fixed time: fluid pressure is constant along (moving) boundary.

- $SL(n)$ is a subgroup of the group of volume preserving diffeos of $\mathbb{R}^n$.
- Boundary conditions satisfied by $SL(n)$ geodesic $\iff$ under the Riemannian metric induced by the geometry of $\Omega$, $SL(n)$ is totally geodesic.
**Theorem**. This holds iff $\Omega$ is a round ball (and hence $I_\Omega = \mathrm{Id}_n$).

\(A \to A^T\) is isometry of \(M(n)\), and fixes \(SL(n)\), so is isometry on \(SL(n)\).

**Lemma**. $S(n)\cap SL(n)$ is totally geodesic in \(SL(n)\).

**Corollary**. $\mathring{S}_+(n)$ is totally geodesic in \(SL(n)\).

- \(SO(n)\) acting on \(M(n)\) by matrix multiplication is isometric & fixes \(SL(n)\)
- Distinct left and right actions!
- Generate a total of \(n(n-1)\) Killing vector fields
__Physical interpretation:__$\quad$*angular momentum*and*vorticity*

- Conjugate action: $A \mapsto OAO^{-1}$ (fixes $\mathring{S}_+(n)$)
- Composite action: $A \mapsto OAO$ (not faithful)

Every $A\in SL(n)$ has a *unique* factorization as $OP$ (or $PO$): $
\begin{cases} O\in SO(n) \newline P\in \mathring{S}_+(n)\end{cases}$

$SL(n)$ foliated by cosets

*So the geodesics are easy to describe, right?*

**Problem**: Left actions generate right-invariant vector fields and vice versa

... so even with our conserved quantities, the motions on the two factors do not split.

- Motion is completely integrable: $\dim(SL(2)) = 3$ and conservation of
- Energy
- Angular momentum
- Vorticity

- Used in Roberts-Shkoller-Sideris (CMP 2020) where geodesics are fully classified.

Will return to this classification a bit later.

- $SL(2)$ is a
*warped product*$\mathring{S}_+(2) \times_f \mathbb{S}^1$ - Use the composite $A\mapsto OAO$ action
*double cover*of $SL(2)$ by $\mathring{S}_+(2) \times_f SO(2)$.- Use $SO(2) = U(1)$ and "take square roots"

- $SO(2)$ is abelian:
- Composite $OAO$ action is orthogonal to cosets of polar decomposition
- Conjugate $OAO^{-1}$ action is tangent to cosets

- Decompose as integrable motion on $\mathring{S}_+(2)$ plus fibre motion.
- (Private communication from Ben Schmidt)

Call $SO(n)\subsetneq SL(n)$ the "throat": it describes the points closest to the origin in $M(n)$.

Sign convention: $II(X,X) \gt 0$ if it curves away from the origin.

Second f.f. defined via normal pointing away from origin of \(M(n)\).

Let $X,Y\in T_A SL(n)$, then \[ II(X,Y) = \frac{\mathrm{tr}(A^{-1}X A^{-1}Y)}{|A^{-1}|} \] and the geodesic equation reads \[ \ddot{A} = \frac{ \mathrm{tr}( A^{-1} \dot{A} A^{-1} \dot{A})}{\mathrm{tr}(A^{-1} A^{-T})} A^{-T}.\]

- The only bounded geodesic orbits the throat
- There exists semi-bounded geodesics converging to the throat
- All unbounded geodesics are "asymptotically linear" ($II(\dot{\gamma},\dot{\gamma}) \to 0$)

How much of these survive in higher dimensions?

Analyses of some special explicit solutions and their asymptotics

Sideris (ARMA 2017)

**Virial Identity**: let $A: t\to M(n)$ a $SL(n)$-geodesic \[ \frac{d^2}{dt^2} |A|^2 = \frac{|\dot{A}|^2}{|A|} + \frac{n}{|A|^2} II(\dot{A}, \dot{A}).\]**Proposition**: if additionally $A$ is such that $II(\dot{A}, \dot{A}) \gt 0$ for all $t \gt t_0$, then $A$ is unbounded and asymptotically linear.

Sideris (ARMA 2017)

**Theorem.**$SL(2)$ has bounded sectional curvature.**Theorem.**When $n \geq 3$, given $A\in SL(n)$ and $W\subset T_A SL(n)$ a plane, denote by $k(W)$ its sectional curvature. Then:- $k(W) \leq |A^{-1}|^2$ for all $(A,W)$;
- There exists a sequence $(A_i, W_i)$ with

$|A_i^{-1}| \to \infty$ and $\liminf k(W_i) |A_i^{-1}|^{-2} \geq \frac14$.

*Remark*: the function $A\mapsto |A^{-1}|$ is proper.)

**Linear solutions**: those whose images are lines in $M(n)$.

All in the form on $A(t) = A_0(\mathrm{Id} + t M)$ where $M$ nilpotent.**Linear stability**: Jacobi fields along such curves grow at most linearly.- Key step: showing for such $A(t)$ the quantity \[ \left| \mathrm{Riem}(\dot{A}, J, \dot{A}, K) \right| \lesssim \frac{|J|~ |K|}{(1 + t^4)}. \]

**Theorem.**When $n = 2m \geq 4$, for every $\eta \geq |\mathrm{Id}|$, there exists a bounded geodesic $A(t)$ with $|A(t)| \equiv \eta$.- In fact, we can write $A(t) = A_0 \exp (Ct)$ with $A_0 \in \mathring{S}_+(n)$ and $C\in \mathfrak{so}(n)$.
*Remark*: these are the only "non-linear" "exponential" solutions.*Physical picture:*Stationary fluid solution; an ellipsoid with "paired" axes, spinning separately in each plane.- When the rotational speeds are not "tuned", the solutions remain bounded, but pulsates.

\[ \frac{d^2}{dt^2} |A|^2 = \frac{|\dot{A}|^2}{|A|} + \frac{n}{|A|^2} II(\dot{A}, \dot{A}).\]

**Lemma**. When $n = 2$, we have $\frac{d^2}{dt^2} |A(t)|^2 \leq 0 \implies |A(t)| = |\mathrm{Id}|$.

**Corollary**. All geodesics except for the throat rotation are unbounded.

Emphatically false when $n \geq 3$.

**Theorem.** [sufficient conditions for unboundedness]

- Solution is tangent to $\mathring{S}_+(n)$ (or another coset)
- Solution has vanishing total angular momentum
- Solution has vanishing total vorticity

Given solution $A(t)$, define \[ \beta = A^{-1}A^{-T}, \quad \omega = A^T \dot{A} + \dot{A}^T A, \quad \zeta = A^T \dot{A} - \dot{A}^T A \]

- $\beta$: "square" of the "radial component" in polar decomposition
- $\omega$: "radial" velocity
- $\zeta$: "angular" velocity

Above is the vorticity version; angular momentum version moves the transpose to the other factor.

Cannot do both simultaneously.

*Solves* problem with Killing vector field not pointing in the fibre direction.

\begin{align*} \frac{d}{dt} \beta & = -\beta \omega \beta \newline \frac{d}{dt}\omega & = \frac12(\omega + \zeta)^T\beta(\omega + \zeta) + \underbrace{\frac{\mathrm{tr}\omega\beta\omega\beta}{2\mathrm{tr}\beta}}_{\geq 0} I + \underbrace{\frac{\mathrm{tr}\zeta\beta\zeta\beta}{2\mathrm{tr}\beta}}_{\leq 0} I \newline \frac{d}{dt}\zeta &= 0 \end{align*}

Compatibility condition: $\quad \mathrm{tr}\beta\omega = 0$.

Preserves "block diagonal structure"

Assume $\beta, \omega,\zeta$ decompose into $2\times 2$ blocks (with one $1\times 1$ block when $n$ is odd) along the diagonal.

**Theorem**. Let $n$ be even. If the initial data of $\beta,\omega$ are built from $2\times 2$ blocks that are pure-trace, and if $\zeta$ has no vanishing bocks, then the solution is bounded.

When $n = 2$, hypothesis only possible with rotation at throat.

**Theorem**. If the initial data has the form
\[ \beta_0 = \begin{pmatrix} b_1 \mathrm{Id}_{2m} \newline & b_2 \mathrm{Id}_{n - 2m}\end{pmatrix}, \quad \omega_0 = \begin{pmatrix} w_1 \mathrm{Id}_{2m} \newline & w_2 \mathrm{Id}_{n-2m} \end{pmatrix} \]
and writing $\varepsilon$ for the antisymmetric $2\times 2$ matrix
\[ \zeta_0 = \begin{pmatrix} z (\underbrace{\varepsilon \oplus \cdots \oplus \varepsilon}_{m \text{ copies}}) \newline & 0 \end{pmatrix};\]
then the corresponding solution is unbounded, and asymptotically linear.

Sideris [ARMA 2017] described $n = 3$ and $m = 1$ with explicit integration

- Let $A(t)$ be a swirling and shear flow: we can show that $II(\dot{A}, \dot{A})$ is eventually positive.
- So $A(t)$ is "asymptotically linear"
- "Small perturbations" \[ \zeta_0 = \begin{pmatrix} z (\underbrace{\varepsilon \oplus \cdots \oplus \varepsilon}_{m \text{ copies}}) \newline & 0 \end{pmatrix} \Longrightarrow \begin{pmatrix} z (\underbrace{\varepsilon \oplus \cdots \oplus \varepsilon}_{m \text{ copies}}) \newline & \delta (\varepsilon \oplus \cdots \oplus \varepsilon) \end{pmatrix} \]
- Non-vanishing rotation $\implies$ motion is bounded

- Bounded motions must enter regime with negative second fundamental form.
*Physical instability*: Rayleigh-Taylor. (Require pressure just inside the boundary be positive.)

*Conjecture*. When $n$ is odd, $SL(n)$ has**no**bounded geodesics.*Conjecture*. The only solutions with $|A(t)|$ constant are those we found.*Question*. Do there exist unbounded geodesics which are not asymptotically linear when $n \gt 2$?*Question*. Do there exist "heteroclinic orbits"?