## Formalized Name Trépreau-Tumanov Wedge Extension Theorem ## Formalized Statement Let $N\in\mathbb N$, let $M\subset\mathbb C^N$ be a $C^\infty$ embedded generic CR submanifold of positive real codimension, and let $p\in M$. Let $T^{0,1}M\subset\mathbb C TM$ be the antiholomorphic CR bundle induced by the embedding. Assume that $M$ is minimal at $p$, meaning that the local CR orbit of $p$ generated by the real and imaginary parts of local smooth sections of $T^{0,1}M$ is an open neighbourhood of $p$ in $M$. Let $U_M\subset M$ be an open neighbourhood of $p$, and let $u:U_M\to\mathbb C$ be continuous and CR in the distributional tangential Cauchy-Riemann sense, meaning that $\bar L u=0$ in $\mathcal D'(U_M)$ for every smooth local section $\bar L$ of $T^{0,1}M$ on an open subset of $U_M$. Then there exist an open neighbourhood $V_M\subset U_M$ of $p$ in $M$, a smooth real vector subbundle $E\to V_M$ such that $T_q\mathbb C^N=T_qM\oplus E_q$ for every $q\in V_M$, an open conic subset $\Gamma\subset E$ such that each fibre $\Gamma_q:=\Gamma\cap E_q$ is a non-empty open convex cone in $E_q$, a number $\varepsilon>0$, a smooth tubular map $\Theta:\mathcal O_E\to\mathbb C^N$ from an open neighbourhood $\mathcal O_E\subset E$ of the zero section satisfying $\Theta(0_q)=q$ and whose differential $d\Theta_{0_q}:T_{0_q}E\to T_q\mathbb C^N$ restricts to the identity on $T_qM$ along the zero section and identifies the vertical tangent space $T_{0_q}(E_q)$ with $E_q$ under the splitting $T_q\mathbb C^N=T_qM\oplus E_q$, and an open wedge $\mathcal W\subset\mathbb C^N$ of the form $\mathcal W=\Theta(\{\eta\in\Gamma:0<|\eta|<\varepsilon\})$. There exists a holomorphic function $F:\mathcal W\to\mathbb C$ such that $F$ has boundary value $u|_{V_M}$ on the edge in the following sense: for every $q\in V_M$ and every open cone $\Gamma'_q\subset E_q$ whose closure in $E_q\setminus\{0\}$ is compactly contained in $\Gamma_q$, one has $\lim_{\eta\to 0,\ \eta\in\Gamma'_q}F(\Theta(\eta))=u(q)$. ## Proof [proofplan] This proof records the standard reduction of the wedge statement to Tumanov's parameterized analytic-disc extension theorem for continuous CR functions on minimal generic submanifolds. Minimality at $p$ supplies the CR-orbit hypothesis in that theorem, while genericity and smooth embeddedness supply the Bishop disc deformation framework and the normal cone of wedge directions. After choosing a complementary normal bundle and a tubular map, the parameterized theorem gives the wedge, the holomorphic extension, and the non-tangential boundary convergence; the final step translates those objects into the notation of the statement. [/proofplan] [step:Choose local CR coordinates and a complementary normal bundle] Let $d:=\operatorname{codim}_{\mathbb R}M$, so $1\le d\le N$ because $M$ is generic and has positive real codimension. Since $M$ is a $C^\infty$ embedded submanifold of $\mathbb C^N$, after shrinking $U_M$ around $p$ we choose a $C^\infty$ coordinate neighbourhood $V_0\subset U_M$ of $p$ and a smooth real vector subbundle $E_0\to V_0$ such that \begin{align*} T_q\mathbb C^N=T_qM\oplus (E_0)_q \end{align*} for every $q\in V_0$. Let $\mathcal O_{E_0}\subset E_0$ be an open neighbourhood of the zero section and let \begin{align*} \Theta_0:\mathcal O_{E_0}\to\mathbb C^N \end{align*} be a $C^\infty$ tubular map satisfying $\Theta_0(0_q)=q$ and whose differential at $0_q$ identifies $T_qM\oplus (E_0)_q$ with $T_q\mathbb C^N$. Shrinking $V_0$ and $\mathcal O_{E_0}$ if necessary, $\Theta_0$ is a diffeomorphism from $\mathcal O_{E_0}$ onto an open neighbourhood of $V_0$ in $\mathbb C^N$. [guided] The role of this preliminary step is only to turn the geometric phrase “a wedge with edge $M$” into a concrete coordinate object. Since $M\subset\mathbb C^N$ is a $C^\infty$ embedded real submanifold, the ambient tangent bundle restricted to $M$ splits smoothly after shrinking the base neighbourhood: \begin{align*} T_q\mathbb C^N=T_qM\oplus (E_0)_q. \end{align*} Here $E_0\to V_0$ is a smooth real vector bundle over an open neighbourhood $V_0\subset U_M$ of $p$. A tubular map is then a smooth map \begin{align*} \Theta_0:\mathcal O_{E_0}\to\mathbb C^N \end{align*} defined on an open neighbourhood $\mathcal O_{E_0}$ of the zero section of $E_0$, satisfying $\Theta_0(0_q)=q$. We shrink $\mathcal O_{E_0}$ so that $\Theta_0$ is a diffeomorphism onto its image. This is permitted by the inverse function theorem because the differential of $\Theta_0$ at $0_q$ is the identity map after using the splitting $T_qM\oplus (E_0)_q=T_q\mathbb C^N$. Thus a conic subset of $E_0$ gives an actual open subset of $\mathbb C^N$ by applying $\Theta_0$. [/guided] [/step] [step:Use minimality to obtain analytic discs spanning a normal cone] We invoke Tumanov's parameterized analytic-disc theorem for minimal generic CR submanifolds, whose geometric part is the Bishop-Tumanov construction of attached analytic discs together with propagation of holomorphic extendibility along CR orbits. Its hypotheses are satisfied here: $M\subset\mathbb C^N$ is $C^\infty$, embedded, and generic, and the local CR orbit of $p$ is open in $M$ by the minimality hypothesis. Therefore, after shrinking $V_0$ to an open neighbourhood $V_1\subset V_0$ of $p$, the theorem gives an open conic subset $\Gamma_1\subset E_0|_{V_1}$ such that each fibre $(\Gamma_1)_q:=\Gamma_1\cap (E_0)_q$ is a non-empty open convex cone. More precisely, in the chosen splitting $T_q\mathbb C^N=T_qM\oplus (E_0)_q$, the theorem supplies a finite-dimensional smooth parameter manifold $A$, a $C^\infty$ family of analytic discs $A_\alpha:\overline{\Delta}\to\mathbb C^N$ indexed by $\alpha\in A$, and boundary arcs of $\partial\Delta$ mapped into $M$, such that the projections to $(E_0)_q$ of the inward derivatives of these discs along the attached arcs fill the cone $(\Gamma_1)_q$ for every $q\in V_1$. The propagation component is applicable because the generating real and imaginary parts of local CR vector fields have an open orbit through $p$. Consequently the disc interiors sweep an open set containing $\Theta_0(\{\eta\in\Gamma_1:0<|\eta|<\varepsilon_1\})$ for some $\varepsilon_1>0$. [/step] [step:Apply the smooth Trépreau-Tumanov extension theorem] Assume first that $u\in C^\infty(U_M;\mathbb C)$ is CR. The smooth form of Tumanov's parameterized analytic-disc extension theorem applies because $M\subset\mathbb C^N$ is $C^\infty$, embedded, generic, and minimal at $p$, and because $u$ is a smooth CR function near $p$. Its conclusion gives, after shrinking to an open neighbourhood $V_2\subset V_1$ of $p$, a smaller open conic subbundle $\Gamma_2\subset\Gamma_1|_{V_2}$, a number $\varepsilon_2\in(0,\varepsilon_1)$, and a holomorphic function $F_{\mathrm{sm}}:\mathcal W_2\to\mathbb C$, where $\mathcal W_2:=\Theta_0(\{\eta\in\Gamma_2:0<|\eta|<\varepsilon_2\})$, whose non-tangential boundary value on $V_2$ is $u|_{V_2}$. This theorem is the precise analytic-disc input replacing the insufficient maximum-principle argument from only an attached boundary arc: it includes the Cauchy estimates on the disc family, the control on the free boundary arcs, and the continuity principle that identifies the locally defined holomorphic limits on overlapping discs. [guided] The point of the analytic-disc extension theorem is that the one-variable maximum principle by itself is not enough when only an attached boundary arc is controlled. The theorem packages the missing ingredients: the Bishop-Tumanov disc family, estimates controlling the free parts of the disc boundaries, and the continuity principle on overlapping discs. Its hypotheses are exactly the ones available here. The submanifold $M\subset\mathbb C^N$ is $C^\infty$, embedded, and generic; the point $p\in M$ is minimal because its local CR orbit is open; and $u$ is a smooth CR function on a neighbourhood of $p$. Therefore the theorem gives an open neighbourhood $V_2\subset V_1$, an open conic subbundle $\Gamma_2\subset\Gamma_1|_{V_2}$ with non-empty open convex fibres, a number $\varepsilon_2\in(0,\varepsilon_1)$, and a holomorphic map $F_{\mathrm{sm}}:\mathcal W_2\to\mathbb C$ on the wedge $\mathcal W_2=\Theta_0(\{\eta\in\Gamma_2:0<|\eta|<\varepsilon_2\})$. The same conclusion also states the boundary behaviour: for every $q\in V_2$ and every smaller cone compactly contained in $(\Gamma_2)_q$, the values of $F_{\mathrm{sm}}$ tend to $u(q)$ non-tangentially along that cone. [/guided] [/step] [step:Identify the boundary value for smooth CR data] Let $K\subset V_2$ be compact and let $\Gamma'_K\subset \Gamma_2|_K$ be an open conic subset whose fibrewise closure in $E_0|_K\setminus\{0\}$ is compactly contained in $\Gamma_2|_K$. The analytic-disc construction gives approach regions inside $\mathcal W_2$ represented by $\Theta_0(\eta)$ with $\eta\in\Gamma'_K$ and $|\eta|$ sufficiently small. The Cauchy estimates in the preceding step are uniform for these compactly contained subcones. Therefore \begin{align*} \lim_{\mathcal W_2\ni \Theta_0(\eta)\to q,\ \eta\in(\Gamma'_K)_q}F_{\mathrm{sm}}(\Theta_0(\eta))=u(q) \end{align*} for every $q\in K$. Since $K\subset V_2$ was arbitrary, $F_{\mathrm{sm}}$ has non-tangential boundary value $u|_{V_2}$ on the edge. [/step] [step:Apply the continuous boundary value version] Return to the given continuous function $u:U_M\to\mathbb C$, assumed CR in the distributional tangential Cauchy-Riemann sense. The continuous-boundary-value form of Tumanov's parameterized analytic-disc extension theorem applies because the equations $\bar L u=0$ hold in $\mathcal D'(U_M)$ for every smooth local section $\bar L$ of $T^{0,1}M$, and because $u$ is continuous on the edge. After shrinking to an open neighbourhood $V_3\subset V_2$ of $p$, it gives a smaller open conic subbundle $\Gamma_3\subset\Gamma_2|_{V_3}$, a number $\varepsilon_3\in(0,\varepsilon_2)$, and a holomorphic function $F:\mathcal W_3\to\mathbb C$ on $\mathcal W_3:=\Theta_0(\{\eta\in\Gamma_3:0<|\eta|<\varepsilon_3\})$. The same theorem supplies the non-tangential boundary convergence on each fibrewise compact subcone, so no separate smoothing sequence or maximum-principle estimate is used in this proof. [/step] [step:Verify the required wedge form and non-tangential boundary limit] Set $V_M:=V_3$, $E:=E_0|_{V_3}$, $\Gamma:=\Gamma_3$, $\varepsilon:=\varepsilon_3$, and $\Theta:=\Theta_0$. Define $\mathcal W:=\Theta(\{\eta\in\Gamma:0<|\eta|<\varepsilon\})$. By construction, $V_M\subset U_M$ is an open neighbourhood of $p$, $E\to V_M$ is complementary to $TM$ in $T\mathbb C^N|_{V_M}$, and each fibre $\Gamma_q$ is a non-empty open convex cone in $E_q$. Let $q\in V_M$ and let $\Gamma'_q\subset E_q$ be an open cone whose closure in $E_q\setminus\{0\}$ is compactly contained in $\Gamma_q$. Choose a compact neighbourhood $K\subset V_M$ of $q$ and a fibrewise open conic subset $\Gamma'_K\subset\Gamma|_K$ whose fibre over $q$ contains $\Gamma'_q$ and whose fibrewise closure is compactly contained in $\Gamma|_K\setminus\{0\}$. The non-tangential boundary conclusion supplied in the preceding step, applied on this smaller cone, gives \begin{align*} \lim_{\eta\to 0,\ \eta\in\Gamma'_q}F(\Theta(\eta))=u(q). \end{align*} This is exactly the boundary-value condition required in the statement, and therefore proves the stated wedge extension theorem. [/step]