Immersed Adjunction Inequality in 4-Manifolds

• Immersed adjunction inequality is a topological bound linking the Euler characteristics of ramified tangent and normal bundles with the first Chern number of the pull-back complex structure.
• It employs the canonical Weyl connection and twistor geometry to extend classical minimal surface theory from Kähler to almost-Hermitian manifolds.
• The inequality characterizes equality cases via holomorphic immersions, revealing deep connections between surface topology and the complex geometry of the ambient space.

The immersed adjunction inequality provides a topological bound for branched Weyl-minimal immersions of closed oriented surfaces into conformal $4$–manifolds equipped with an almost-Hermitian structure and their canonical Weyl connection. Generalizing classical results for minimal surfaces in Kähler manifolds, this inequality establishes a precise relationship between the topological invariants of immersed surfaces and the complex geometry of the ambient manifold, with fine structure induced by the choice of the almost-complex structure.

1. Statement and Interpretation of the Inequality

Let $f:\Sigma\to M$ denote a weakly-conformal branched Weyl-minimal immersion of a closed oriented surface $\Sigma$ into a conformal $4$–manifold $(M,c,J)$ equipped with the canonical Weyl connection $D$. The immersed adjunction inequality asserts: $$\chi\bigl(T_f\Sigma\bigr) + \chi\bigl(N_f\Sigma\bigr) \leq \pm c_1\bigl(f^*T^{(1,0)}M\bigr)$$ where $\chi(\cdot)$ is the topological Euler characteristic of the (ramified) tangent bundle $T_f\Sigma$ and its orthogonal complement $N_f\Sigma$; $f^*T^{(1,0)}M$ is the pull-back of the $J$-holomorphic tangent distribution of $M$, and $c_1(\cdot)$ refers to its first Chern number. The sign is determined by the orientation choice: the "+" sign for the almost-complex structure $J$, and "–" for its conjugate $-J$.

Branch points—isolated singularities where $df$ drops rank—are intrinsic to the branched setting, and the inequality remains valid for immersions with such singularities. The inequality provides a conformally invariant analogue of classical adjunction bounds for minimal immersions in standard Hermitian geometry (Ream, 2019).

2. Fundamental Definitions and Structures

The surface $\Sigma$ inherits the induced conformal class $f^*c$, ramified tangent bundle $T_f\Sigma\subset f^*TM$, and orthogonal normal bundle $N_f\Sigma$. The ambient manifold $(M, c, J)$ is almost-Hermitian: $c$ a conformal class of metrics, $J$ an almost-complex structure. The canonical Weyl connection $D$ is a torsion-free connection preserving $c$, equivalent to a connection on the density bundle $L \rightarrow M$.

Given a local metric $g\in c$, the Weyl connection is realized as: $$\nabla^{D}_X Y = \nabla^{g}_X Y + \alpha_g(X)Y + \alpha_g(Y)X - g(X,Y)\alpha_g^\sharp$$ where $\alpha_g$ is the Lee form capturing the deviation of $D$ from the Levi-Civita connection. The Weyl second fundamental form $B$ for an immersion $f:\Sigma\to M$ is given by: $B = A_g - (\alpha_g^\sharp)^\perp \otimes g$ with $A_g$ the classical second fundamental form; $f$ is Weyl-minimal if $\mathrm{trace}_{f^*g} B = 0$. This condition is conformally invariant and specializes to classical minimality if $D$ is exact.

The $J$–holomorphic tangent bundle $T^{(1,0)}M$ is the subbundle of $TM\otimes \mathbb C$ spanned by vectors of type $(1,0)$ for the almost-complex structure $J$. Pulling this back via $f$ yields a complex line bundle on $\Sigma$ whose first Chern number is a key term in the inequality.

3. Twistor Geometry and Weyl-Minimality

The weightless twistor space $\mathcal{Z}_\pm \rightarrow M$ consists of the $S^2$–bundle of self-dual ($+$) or anti-self-dual ($-$) 2-forms of unit length in the conformal metric. Using the Weyl connection $D$, one constructs almost-complex structures $\mathcal{J}_\pm$ on $\mathcal{Z}_\pm$. The Eells–Salamon-type correspondence furnishes a geometric representation:

"A weakly-conformal map $f:\Sigma\to M$ is Weyl-harmonic (i.e., Weyl-minimal and branched at isolated points) if and only if its twistor lift $\tilde f_\pm :\Sigma\to\mathcal Z_\pm$ is a non-vertical $\mathcal J_\pm$–holomorphic curve." (Ream, 2019)

This correspondence bridges the analytic properties of Weyl-minimal surfaces with holomorphic curve theory in twistor spaces, thus exporting topological inequalities into the twistor framework.

4. Sketch of Proof and Combinatorial Structure

Fixing a local holomorphic coordinate $z$ on $\Sigma$, with $\partial_{\bar z}$ spanning the antiholomorphic tangent, one expands: $$f_z = \alpha + \bar\beta, \qquad \alpha\in f^*T^{(1,0)}M, \quad \bar\beta\in f^*T^{(0,1)}M$$ with $$\langle\alpha,\alpha\rangle = \langle\beta,\beta\rangle = 0$$. The Weyl-harmonic equation $\nabla^D_{\bar z}f_z = 0$ and properties of the canonical Weyl connection ($d^D\omega_c=0$, $\nabla^D J$ $J$–anti-linear) yield: $$\bar\partial\,\alpha = \phi\, \alpha\otimes d\bar z, \quad \bar\partial\,\bar\beta = \psi\, \bar\beta\otimes d\bar z$$ where $\phi$ and $\psi$ involve the Nijenhuis tensor $N$ of $J$.

Applying the Bers–Vekua similarity principle, zeros of $\alpha$ and $\bar\beta$ have positive integer order; the total ramification $R$, the number of anti-complex points $Q$, and complex points $P$ are thus well-defined non-negative integers. The Riemann–Roch theorem and Koszul–Malgrange formulas provide

$$\chi(T_f\Sigma)+\chi(N_f\Sigma) = -(P+Q), \qquad c_1\bigl(f^*T^{(1,0)}M\bigr) = Q-P$$

Since $P, Q \ge 0$, the adjunction bound follows.

5. Characterization of Equality and Holomorphic Cases

The inequality becomes an equality precisely when either $P$ or $Q$ vanishes. $P=0$ (respectively $Q=0$) indicates absence of positively oriented complex points (respectively anti-complex points), so $f$ is $J$–holomorphic (respectively $-J$-holomorphic). These $\pm J$–holomorphic curves are Weyl-minimal for the canonical Weyl connection and realize the adjunction bound with equality.

Such holomorphic curves are thus extremal for the immersed adjunction inequality, and their existence is tightly linked to the underlying complex and conformal geometry of $(M,c,J)$.

6. Relation to Classical Theory and Generalizations

If $D$ is exact—i.e., $(M,c)$ has a globally defined metric within the conformal class—Weyl-minimality reduces to classical minimality. The immersed adjunction inequality then recovers the well-known bounds of Eells–Salamon and Webster for minimal surfaces in Kähler $4$–manifolds.

In the almost-Kähler regime (closed Lee form, so $D$ exact on the universal cover), the result extends prior work by Chen–Tian, Ville, and Ma to the context of branched immersions. More generally, the inequality establishes that topological constraints on immersed minimal surfaces in $4$–manifolds admit a conformally invariant form by substituting Weyl-minimality and leveraging the canonical Weyl connection of an almost-Hermitian structure.

A plausible implication is the broader applicability of these topological bounds to settings beyond globally Kähler manifolds, hinging on conformal and almost-complex structures rather than strict integrability or metric exactness.

7. Topological and Geometric Consequences

The immersed adjunction inequality yields immediate global restrictions on the topology of Weyl-minimal surfaces in $4$–manifolds:

• It constrains the possible values of the Euler characteristic of ramified tangent and normal bundles relative to the geometry of the ambient manifold via its complex Chern numbers.
• The bound is sharp for holomorphic immersions, echoing adjunction-type results elsewhere in complex geometry.
• This suggests new avenues of inquiry in conformally invariant surface theory and opens connections to the study of holomorphic curves in generalized twistor spaces.

The established correspondence between branched Weyl-minimal surfaces and twistor holomorphic curves provides a geometric mechanism for translating analytic conditions into topological inequalities, generalizing the reach of adjunction principles across the broader category of almost-Hermitian $4$–manifolds (Ream, 2019).