Cartan uniqueness theorem on nonopen sets

Theorem: (Cartan) Suppose $U \subset \C^n$ is a bounded domain, $p \in U$, and $f \colon U \to U$ is holomorphic such that $f(p)=p$ and $Df(p) = I$. Then $f$ is the identity map.

The proof is to iterate and note that the first nonzero higher order derivative would grow without bound, while it must also stay bounded by Cauchy estimates.

We wish to generalize to nonopen sets.

For any set $V \subset \C^n$, the set of functions we consider, $\sO(V),$ are restrictions of functions holomorphic in some neighborhood of $V$.

The theorem does not hold for a submanifold or a subvariety in general.

E.g., it does not hold for the sphere in $\C^n$.

So we wish to figure out for what sets does it hold.

We also want a local analogue to investigate automorphisms on germ level.

For the main result we will not require much for the set.

It could be a manifold, a variety, or really any locally closed, locally connected set.

Cartan's theorem is really a jet determination result.

Chern-Moser '74 proved that for a real-analytic Levi-nondegenerate hypersurface, automorphisms are determined by the 2-jet.

Jet determination has been much studied since then (see e.g. the surveys by Zaitsev or Baouendi-Ebenfelt-Rothschild).

Cartan's theorem is then 1-jet determination.

We will prove a Cartan-like theorem under two conditions.

First, we need to extend $\sO(V)$ functions into a fixed neighborhood where we can iterate.

Analytic discs are good for this. How about the disc hull:

Given compact $K \subset \C^n$, let $\hat{K}_D$ be the set of points $p$ such that for every $\epsilon > 0$, there is an analytic disc $\varphi \colon \overline{\D} \to \C^n$ such that $\varphi(0) = p$ and $\varphi(\partial \D)$ in an $\epsilon$-neighborhood of $K$.

Disc hulls are used to construct the polynomial hull of a set. (See e.g., Poletsky '93 or Porten '17)

Unfortunately if $\bT \subset \C$ is the unit circle, then $\hat{\bT}_D = \overline{\D}$, and not every function in $\sO(\bT)$ extends holomorphically to $\D$.

We want to use the Kontinuitätssatz, so we need families of discs along which to extend.

So we define a contracting disc hull:

Given compact $K \subset \C^n$, let $\hat{K}_{CD}$ be the set of points $p$ such that for every $\epsilon > 0$, there is an one dimensional family ($t \in [0,1]$) of analytic discs $\varphi_t \colon \overline{\D} \to \C^n$ such that $\varphi_1(0) = p$ and $\varphi_t(\partial \D)$ in an $\epsilon$-neighborhood of $K$ for all $t \in [0,1]$ and $\varphi_0(\overline{\D})$ is in an $\epsilon$-neighborhood of $K$.

Such families of discs are used to construct the envelope of holomorphy of a neighborhood of $K$.

A set $V$ satisfies the contracting disc hull condition at $p \in V$ if for every compact neighborhood $K \subset V$ of $p$, $\hat{K}_{CD}$ contains $p$ in its interior.

Contracting disc hull condition guarantees a unique extension to a fixed neighborhood.

What about the derivative being the identity?

If $V$ is not open, then for $f \in \sO(V)$, the derivative $Df(p)$ is not well-defined on $T_p\C^n$.

It is well-defined on the tangent cone $C_p V$:

The set of vectors $v$ that are limits of sequences $r_j (q_j-p)$ where $r_j > 0$ and $q_j \in V$.

So when does $Df(p)$ being the identity on $C_p V$ imply $Df(p)=I$?

When $C_p V$ is not contained in any complex subspace, that is, if the complex span of $C_p V$ is $T_p \C^n$.

Theorem: Let $V \subset \C^n$ be a connected, bounded, locally connected, and locally closed set, and let $p \in V$. Suppose that $V$ satisfies the contracting disc hull condition at $p$, and suppose that the complex span of $C_p V$ is $T_p \C^n$. Let $f \colon V \to V$ be a mapping in $\sO(V)$ such that $f(p) = p$ and the derivative $Df(p)$ restricted to $C_p V$ is the identity. Then $f(z) = z$ for all $z \in V$.

Example: $V \subset \C^3$ given by $\operatorname{Im} z_3 = |z_1|^2-|z_2|^2$ and $\|z\| < 1$. Satisfies both conditions. Theorem holds.

Example: $V \subset \C^6$ given by $\operatorname{Im} z_3 = |z_1|^2-|z_2|^2$, $\operatorname{Im} z_6 = |z_4|^2-|z_5|^2$, and $\|z\| < 1$. Satisfies both conditions. Theorem holds.

Example: Interestingly, the genericity of $C_p V$ is not always necessary. Let $V$ be as above.

If $\Phi$ is a map, $\Phi(0)=0$, $D\Phi(0)$ noninvertible, such that $\Phi|_V$ is a diffeomorphism, then $\Phi(V)$ is a CR singular submanifold and $C_0 \Phi(V) = T_0 \Phi(V)$ not generic.

But as the theorem is satisfied on $V$ it is satisfied on $\Phi(V)$.

If $\Phi$ is finite, then $\Phi(V)$ satisfies the contracting disc hull condition.

Example: The contracting disc condition on its own is not enough to give the theorem however.

$V$ given by \[ |z_2| \leq |z_1|^2, \qquad |z_1| \leq 1 \] satisfies the contracting disc hull condition.

$C_0 V = \{ z_2 = 0 \}$

$(z_1,z_2) \mapsto (z_1,z_2^2)$ violates the conclusion of the theorem.

Infinitesimal automorphisms.

If $\operatorname{Re} Z$ is a vector field tangent to a manifold for a holomorphic vector field $Z$, then $\operatorname{Re} Z$ is an infinitesimal automorphism.

Generalizes to subvarities.

Theorem: Suppose $V \subset \C^n$ is a bounded local real-analytic subvariety that satisfies the contracting disc hull condition at $p \in V$. Suppose that $X$ is an infinitesimal automorphism whose flow on some neighborhood of $V$ exists for all positive time and is $O(2)$ at $p$. Then $X \equiv 0$.

So "bounded" is replaced by "flow exists for all positive time"

And "derivative is $I$ at $p$" is replaced by "$O(2)$ at $p$."

Thank you for your attention!