[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
JIŘÍ LEBL, ALAN NOELL, AND SIVAGURU RAVISANKAR

Abstract. We study three possible definitions of the notion of CR functions at CR singular points, their extension to a fixed-neighborhood of the singular point, and analogues of the Baouendi–Trèves approximation in a fixed neighborhood. In particular, we give a construction of certain disc hulls, which, if large enough, give the fixed-neighborhood extension and approximation properties. We provide many examples showing the distinctions between the classes and the various properties studied.

Let \( M \subset \mathbb{C}^n \) be a real submanifold and \( T_{\eta}^{0,1}M \) the span of the antiholomorphic vectors at a point \( \eta \in M \). The manifold \( M \) is said to be CR at \( q \) if the dimension of \( T_{\eta}^{0,1}M \) is constant near \( q \). A natural generalization of holomorphic functions is a so-called CR function, a function killed by \( T^{0,1}M \) vector fields. If \( M \) and \( f \) are CR and real-analytic, then \( f \) extends locally to a neighborhood as a holomorphic function by the theorem of Severi [20]. If the regularity is lower, we do not always get such an extension, but basic questions of when and into what set extension holds are relatively well-understood: see the pioneering work by Lewy [15], Kohn–Rossi [9], Tumanov [21], and many others. The problem then is to understand the CR singular setting. In particular, it is not immediately clear what is the most natural notion of CR function. One possible definition is simply to consider functions that are CR at CR points, or equivalently, those that are killed by vector fields valued in \( T_{\eta}^{0,1}M \) at each point. We call these CR functions, and the authors have studied their extension properties in [10, 11, 13].

A related question is the approximation of functions by polynomials or entire functions. In the totally real case, see Hörmander–Wermer [7], Harvey–Wells [5, 6], and in general the survey article by Dwilewicz [3]. The celebrated Baouendi–Trèves theorem [2] applied to CR functions on CR manifolds says that CR functions can be approximated in a fixed neighborhood (not depending on the function) of any point by holomorphic polynomials. At CR singular points, such a theorem does not necessarily hold for the CR functions as we defined them above. We therefore define the class of CR\(_P\) functions to be those that are locally uniform limits of holomorphic polynomials. Mondal [18], extending the work of Mergelyan [16] and Minsker [17], recently studied the approximation property for continuous functions on certain CR singular manifolds that are totally real at CR points. Finally, we write CR\(_H\) for functions that are restrictions of holomorphic functions, that is, those that do

extend to some neighborhood. Sometimes \( \mathcal{O} \) is used for such functions, but we will reserve \( \mathcal{O}(U) \) for functions holomorphic on a specific open set \( U \) for clarity. See Section 2 for precise definitions and the statement of one of our main results.

There are other possible definitions for what constitutes CR functions on a CR singular manifold. For example, Nacinovich–Porten [19] define a class of functions by considering the local closure of germs of \( \text{CR}_H \) functions, and they study its extension properties. Their definition is equivalent to iterating (possibly infinitely many times) an approximation procedure, and hence this class lies in between our \( \text{CR}_H \) and \( \text{CR}_P \).

A subtle but important issue with respect to extension and approximation of CR functions is the size of the neighborhood, namely, whether the neighborhood to which the function extends, or on which it is approximated, depends on the function itself (or not, as is the case for Baouendi–Trèves). For this purpose, we define four different properties a class of functions can satisfy: the extension property and approximation property for functions that extend to be holomorphic to a neighborhood and those that are approximable locally uniformly by holomorphic polynomials, and the fixed-neighborhood extension property and the fixed-neighborhood approximation property if extension or approximation can be done in a neighborhood independent of the function. The extension property implies the approximation property, but not vice-versa. See Section 3 for the definitions and basic examples.

One can use families of discs to extend \( \text{CR}_P \) functions. If we can fill a neighborhood of a point with analytic discs attached to the submanifold, possibly iterating this construction, we can extend \( \text{CR}_P \) functions to this neighborhood. A subtle issue is that we need these discs to be attached to an arbitrarily small neighborhood of a point, that is, we do not a priori have the fixed-neighborhood approximation property for \( \text{CR}_P \) functions at a CR singular point. We formalize this property in Definition 3.2 and define what we call the DH condition.

If we can further shrink such discs to a point, we can apply the Kontinuitätssatz (see Ivashkovich [8]) to get analytic continuation of \( \text{CR}_H \) functions, that is, functions holomorphic in some neighborhood, and we can hope to get a fixed-neighborhood extension property. However, showing that these continuations are single-valued is difficult. For points in the shrinking approximate disc hull, or \( \text{SADH}_q \), we can shrink these discs nicely to the point \( q \). Here we require only approximately attached discs. If \( \text{SADH}_q \) contains a neighborhood of the point \( q \), then we get the fixed-neighborhood extension property (see Corollary 4.4). However, if one requires iteration, much more care has to be taken. We must require the resulting paths in the iteration not to intersect, so we define the iterated SADH condition in Definition 3.4. With this condition, we obtain the fixed-neighborhood extension property in Theorem 4.5.

These ideas can be combined in many different ways with existing results such as the theorem of Hanges and Trèves on the propagation of extension along complex analytic curves through CR points. See Section 4 for these results. In the CR singular case, a large disc hull and in fact the strong iterated SADH condition can appear even in the Levi-flat case, where near the CR points all discs lie in the manifold. In Section 8, we show that the manifold given by \( w = \overline{z}_1 z_2 \), which is Levi-flat at CR points (in fact, an image of \( \mathbb{R}^2 \times \mathbb{C} \)), nevertheless satisfies the strong iterated SADH condition at the origin.

We are interested in producing examples showing that the given classes of functions are distinct. In particular, we wish to extend the Baouendi–Trèves result to a more general CR singular setting. A natural question is whether a class of CR functions on a submanifold of

Cⁿ × ℝ has the approximation property. In this setting, one may guess that the independence of the holomorphic and real coordinates may be used in tandem and that an approximation theorem holds as long as it holds whenever the last coordinate is fixed. That is, perhaps one can combine the classical Weierstrass theorem with Mergelyan’s theorem. It turns out (see Theorem 4.12) that such a result holds with an additional hypothesis. This result shows that fixed-neighborhood extension is not necessary for fixed-neighborhood approximation (see, for example, Section 9). With such a result, one can prove approximation results for flat hyperbolic Bishop surfaces (see Section 7). For flat elliptic Bishop surfaces, in Section 5 we prove approximation under the extra assumption of extension to the natural family of attached analytic discs. For the special elliptic Bishop surface w = |z|², we can adapt the proof of Baouendi–Trèves for the approximation and obtain a linear operator via integration. See Section 6.

Let M ⊂ Cⁿ be a real submanifold of regularity at least C¹. We will assume that all submanifolds are embedded. A point q ∈ M is called a CR point if the dimension of

T_η^{0,1} M = C ⊗ T_η M ∩ span_C { ∂/∂ż₁|_η, ..., ∂/∂ż_n|_η }    (1)

is constant as a function of η in some neighborhood of q. Write M_CR ⊂ M for the set of CR points. A point q ∈ M is said to be a CR singular point if q ∉ M_CR. A submanifold is said to be CR if it has no CR singular points, or in other words if M = M_CR. A vector field L ∈ Γ(C⊗TM) is said to be a CR vector field if L_q ∈ T_q^{0,1} M for all q ∈ M. We remark that our definition of CR vector field includes vector fields at CR singular points as well.

There are several natural definitions of what it means for a function to be CR on a possibly CR singular submanifold. That is, there are different ways to define the analogue of holomorphic functions on a real submanifold. We focus on three such definitions.

Definition 2.1. Suppose M ⊂ Cⁿ is a real C¹ submanifold and f : M → C is a continuous function. We say:

(i) f is a CR function if Lf = 0 for every (continuous) CR vector field L on M, interpreted in terms of distributions if f is only continuous.

(ii) f is a CR_P function if for every q ∈ M there exist a compact neighborhood K ⊂ M of q and a sequence of holomorphic polynomials {P_j} such that P_j converges uniformly on K to f|_K.

(iii) f is a CR_H function if for every q ∈ M there exist a neighborhood U ⊂ Cⁿ of q and a holomorphic function F : U → C such that F|_{M∩U} = f|_{M∩U}.

We write CR^k(M) for the set of CR functions that are in C^k(M), including k = 0 for continuous, k = ∞ for smooth, and k = ω for real-analytic. We define CR_P^k(M) in a similar way. As CR_H functions are always of the same regularity as the manifold, we will write simply CR_H(M).

Remark 2.2. The class CR_H(M) makes sense for any set M, not necessarily a manifold. Correspondingly, some of the basic results in this paper hold without M necessarily being a manifold. Similarly the classes CR_P^k(M) also make sense for a non-manifold M, although the regularity of the functions then needs to be understood, for example, in the sense of Whitney. We leave such generalizations to the reader.

It is easy to see that, for a \( C^1 \) submanifold \( M \), the set of CR points \( M_{CR} \) is an open dense set in \( M \). Therefore, a function \( f \) is CR if and only if \( f|_{M_{CR}} \) is a CR function on \( M_{CR} \). We remark that if \( f \) is a \( CR_P \) function, then \( Lf = 0 \) in the sense of distributions for every CR vector field \( L \), and hence \( f \) is CR. So \( CR_P \) implies CR for all \( C^1 \) submanifolds, but for CR singular submanifolds, the converse may not be true (see below). Note that \( CR_H \) trivially implies CR. In fact, \( CR_H \) implies \( CR_P \) because we can use a series expansion at each point.

When \( M \) is real-analytic and CR, by Severi’s theorem [20] real-analytic CR functions are restrictions of holomorphic functions. That is, in this case \( CR^\omega(M) = CR^\omega_P(M) = CR_H(M) \). On the other hand, there do exist smooth CR functions on CR submanifolds that are not restrictions of holomorphic functions. In the presence of CR singularities, these classes can be distinct even in the real-analytic case.

For CR submanifolds of class \( C^2 \), the first two definitions are equivalent, which follows from the Baouendi–Trèves approximation theorem [2]: If \( M \) is a CR submanifold of class \( C^2 \) and \( q \in M \), then there exists a compact neighborhood \( K \subset M \) of \( q \) such that for every CR function \( f \) on \( M \) there exists a sequence \( \{P_j\} \) of holomorphic polynomials converging uniformly on \( K \) to \( f|_K \). So for a CR submanifold of class \( C^2 \), \( CR^k(M) = CR^k_P(M) \) for all \( k \). A key point in the Baouendi–Trèves theorem is that the neighborhood \( K \) is independent of \( f \); it depends only on \( M \) and the point \( q \). We will see that, in the CR singular case, there exist \( M \) for which the conclusion of the Baouendi–Trèves theorem does not hold even for \( CR_P \) functions. (See Theorem 5.1 or Theorem 9.1.)

When considering regularity of manifolds or functions, we use the order \( 0 < 1 < 2 < \cdots < \infty < \omega \).

## Theorem 2.3.
Let \( M \subset \mathbb{C}^n \) be a real submanifold of regularity \( C^\ell \) for \( \ell \geq 1 \).
(i) \( CR^k(M) \supset CR^k_P(M) \supset CR_H(M) \) for all \( k \leq \ell \).
(ii) There exists a real-analytic submanifold \( M \) such that, for every \( k \), \( CR^k(M) \supsetneqq CR^k_P(M) \).
(iii) There exists a real-analytic submanifold \( M \) such that \( CR^\omega_P(M) \supsetneqq CR_H(M) \).

Proof. (i) This follows from the earlier observations that \( CR_P \) implies CR and \( CR_H \) implies \( CR_P \).
(ii) See Theorem 5.1 or Theorem 8.1.
(iii) See Theorem 7.1. □

The space \( CR_H(M) \) is defined via local extension, but sometimes it is necessary to have one global extension. Recall that a real submanifold is generic at a point if the complex differentials of its defining functions are linearly independent over \( \mathbb{C} \) at that point.

## Lemma 2.4.
Suppose \( M \subset \mathbb{C}^n \) is a \( C^1 \) real submanifold that is either generic at every CR point or simply connected. Then for every \( f \in CR_H(M) \) there exist an open neighborhood \( U \) of \( M \) in \( \mathbb{C}^n \) and a holomorphic function \( F \) on \( U \) such that \( F|_M = f \).

Proof. If \( M \) is generic at every CR point, then the extension is unique locally at each CR point; as those points are dense, the extension is unique everywhere, and the result follows. If \( M \) is simply connected, the result follows by the monodromy theorem. □

## 3. Extension, approximation, and hulls

### Definition 3.1.
Suppose \( M \subset \mathbb{C}^n \) is a real submanifold, \( q \in M \), and \( \mathcal{F} \) is a class of functions on \( M \). We say:

A standard procedure (although it is not sufficient) to construct the polynomial hull is to consider the so-called disc hull. For some sets, we may also have to iterate this procedure, as Example 3.13 shows. Some of the following definitions and examples are stated for an arbitrary set X rather than a submanifold. The reason is that if we iterate the given constructions, we will obtain sets that are not necessarily submanifolds in the intermediate steps.

We make some immediate observations: CR_H(M) always has the extension property and CR^k_P(M) always has the approximation property. The fixed-neighborhood extension property for a class at a point implies the fixed-neighborhood approximation property for that class at that point. If CR^k(M) has the fixed-neighborhood approximation property, then CR^k(M) = CR^k_P(M). These properties are invariant under holomorphic changes of coordinates. If M is contained in the Levi-flat hypersurface given by Im z_n = 0, then CR_H(M) does not have the fixed-neighborhood extension property as 1/(z_n - iε) is in CR_H(M) for all real ε ≠ 0.

Note that the fixed-neighborhood approximation property for CR_H(M) need not imply the fixed-neighborhood extension property for CR_H(M). See any one of Theorems 5.1, 7.1, and 9.1, or the CR case.

When we say simply that M has one of the properties above without mentioning a point q, we mean it has the property at all points. If the submanifold is given, we may say that the class F has the indicated property.

We note that Nacinovich-Porten [19] have studied the extension and approximation properties for a class of functions in between CR_H and CR_P.

Let D ⊂ C denote the unit disc. By an analytic disc attached to X ⊂ C^n we mean a continuous function φ: D → C^n that is holomorphic on D and satisfies φ(∂D) ⊂ X. If we say φ is an analytic disc through p we mean in addition that p ∈ φ(D).

Definition 3.2. Let X ⊂ C^n. Define

DH(X) = {z ∈ C^n : ∃ an analytic disc attached to X through z},     (2)

DH^k(X) = DH(··· DH(DH(X)) ··· )                              (3)
          ⏞   k times   ⏟

We call DH(X) the disc hull of X and DH^k(X) the k-fold iterated disc hull of X.

We say that X satisfies the DH condition at q ∈ X if for every neighborhood U ⊂ X of q, there is a k such that DH^k(U) is a (not necessarily open) neighborhood of q in C^n.

The set DH^k(X) is a subset of the polynomial hull. We wish to apply the Kontinuitätssatz to functions defined on a neighborhood of X, and for this purpose we also need to be able to continuously shrink these discs. It is not always possible to shrink the discs that make up the disc hull even if X is a submanifold (see the examples below), so we require another definition. On the other hand, we do not need to exactly attach these discs, as we only need their boundaries to be near X to apply the Kontinuitätssatz to functions in CR_H. Thus, we can weaken the attachment and only consider approximately attached discs.

Definition 3.3. Let q ∈ X ⊂ C^n. For ε > 0 let X_ε denote the ε-neighborhood of X. Define
SADH_q(X) = {z ∈ C^n : for each ε > 0, ∃ a continuous family of analytic discs φ_t : 𝔻 → C^n,
t ∈ [0, 1], z = φ₁(0), φ_t(∂𝔻) ⊂ X_ε ∀ t ∈ [0, 1], φ₀ ≡ q, and
‖φ_t(0) − q‖ is a strictly increasing function of t},                (4)

We call SADH_q(X) the shrinking approximate disc hull of X at q.

We remark that the content of Definition 3.3 is unchanged if we require z ∈ φ₁(𝔻) instead of φ₁(0) = z.

Finally, we need to take into account single-valuedness of the extension, especially if we are planning to iterate this construction. It is possible to extend functions in CR_H(X) to a ball in SADH_q(X) if it in fact contains such a ball, but it is not possible simply to iterate this procedure. That is, just because SADH_q(SADH_q(X)) contains a ball does not mean we can extend to this ball—we need extra assumptions. One possible way to do this is to ensure that the paths along which we can extend do not create loops. With an eye towards the application to weighted homogeneous submanifolds, and specifically the example in Theorem 8.1, we make the following definition, which will allow iterating the shrinking disc hulls.

Definition 3.4. We say Y is a union of a compact family of nonintersecting paths from SADH_q(X) if there is a family 𝒴 of paths ψ(t) = φ_t(0) arising from the definition of SADH_q(X), 𝒴 is compact with respect to the uniform norm, Y is the union of the images of paths in 𝒴, and whenever ψ₁, ψ₂ ∈ 𝒴 are such that ψ₁(t₁) = ψ₂(t₂) for some t₁, t₂ > 0, then ψ₁([0, t₁]) = ψ₂([0, t₂]).

We will call the sets X₀, X₁, . . . , X_k SADH iterates at q if X₀ ⊂ X is a neighborhood of q, and each X_j for j = 1, . . . , k is a union of a compact family of nonintersecting paths from SADH_q(X_{j−1}).

We say that X satisfies the iterated SADH condition at q if X₀ = X and there are sets X₁, X₂, . . . , X_k that are SADH iterates at q and X_k contains a neighborhood of q in C^n.

We say that X satisfies the strong iterated SADH condition at q if X satisfies the iterated SADH condition at q for all neighborhoods X₀ ⊂ X of q.

We will show that if SADH_q(X) contains a ball centered at q, then functions from CR_H(X) will extend to that ball. However, it is not true that functions in CR_H(X) necessarily extend holomorphically to a neighborhood of SADH_q(X). We can use the Kontinuitätssatz, but that only gives us analytic continuation along the paths arising from the disc families, not unrestricted continuation. Without some condition on SADH_q(X) (or a subset), it is not always possible to have functions in CR_H(X) to extend to CR_H(SADH_q(X)) precisely because we do not get unrestricted continuation. See Example 3.14.

A few remarks are in order.

Remark 3.5. The metric used in the definition of SADH_q(X) need not be the Euclidean distance; other metrics could easily be used.

Remark 3.6. The way we defined SADH_q(X) used paths of strictly increasing distance. We get the same set if we relax this definition simply to increasing distance: If p can be reached by a family of discs where ‖φ_t(0) − q‖ is only increasing and with boundary in an (ε/2)-neighborhood of X, we can perturb the family to get a strictly increasing ‖φ_t(0) − q‖ and boundary in an ε-neighborhood of X. Assuming ‖φ_t(0) − q‖ is strictly increasing makes some of the arguments somewhat cleaner, and as we just saw, there is no loss in generality.

We will generally apply SADH_q to neighborhoods of q in a manifold. A manifold is always locally path connected and, moreover, it is (locally) a union of paths of increasing distance from q ∈ M. So if X is such a neighborhood in a manifold, then X ⊂ SADH_q(X), and SADH_q(X) is (again) a union of paths of increasing distance from q.

Remark 3.7. It is easy to see that SADH_q(X) is contained in the rational hull of X: Consider any rational function f that is holomorphic in a neighborhood of X. By the Kontinuitätssatz, f analytically continues to any point of SADH_q(X), and hence SADH_q(X) does not intersect the pole set.

It is useful to know whether the hulls we define are compact if we are starting with a compact set. The set SADH_q(X) is compact if X is, but unfortunately DH(X) need not be compact. However, the polynomial hull is always compact.

Lemma 3.8. If X ⊂ ℂⁿ is compact, then SADH_q(X) is compact.

Proof. That SADH_q(X) is bounded follows from the maximum principle. To see that it is closed, suppose p₁ ∈ SADH_q(X). For any ε > 0 there is a p₂ ∈ SADH_q(X) that is within ε/2 of p₁. There is an approximately attached shrinking family of discs where the last disc goes through p₂ and the boundaries are within ε/2 of X. By adding a small function linear in t to the family, we can create a new family of discs where the last disc goes through p₁ and the boundaries are within ε of X. As ε > 0 was arbitrary, p₁ ∈ SADH_q(X). □

Example 3.9. Given a compact set X, the set DH(X) need not be compact. Let φ_k : 𝔻 → ℂ, k ∈ ℕ, be a uniformly bounded sequence of analytic functions such that φ_k(0) = 0 and φ_k|_{∂𝔻} converges to a bounded function that is continuous except at the point 1 ∈ ∂𝔻. Assume also that the sequence φ_k converges on 𝔻 \ {1} to a continuous function φ_∞ holomorphic on 𝔻. For simplicity, we could also arrange that each φ_k extends a little past the circle. It is not difficult to construct such a sequence. Define

X = {z ∈ ℂ³ : z₁ ∈ ∂𝔻, z₂ = φ_k(z₁), z₃ = 1/k, k ∈ ℕ}. (5)

Any disc attached to X lies in a set where z₃ is constant, and hence either z₃ = 0 or z₃ = 1/k for some k. For each k ∈ ℕ define Φ_k(ζ) = (ζ, φ_k(ζ), 1/k). The discs Φ_k are all attached to X, and their images lie in DH(X). Moreover, for k ∈ ℕ, let Ψ(ζ) = (α(ζ), β(ζ), 1/k) be an analytic disc. The image of Φ_k is a subvariety, and it is given by functions that are holomorphic a little bit past the boundary. Thus, the analytic disc Ψ takes the circle into this subvariety, and hence it takes 𝔻 into the subvariety. In other words, the image Φ_k(𝔻) gives all the points in DH(X) for z₃ = 1/k. Note that (0, 0, 1/k) ∈ DH(X).

Now suppose that Ψ(ζ) = (α(ζ), β(ζ), 0) is an analytic disc. As |α(ζ)| = 1 for |ζ| = 1, we have that α is a finite Blaschke product. In particular, either it is constant, or it goes

For smooth generic CR submanifolds, local attached analytic discs form a Banach manifold, and all such discs will shrink to a point; see Sections 6.5 and 8.2 of [1]. However, shrinking families of discs are not guaranteed for CR singular submanifolds near the CR singular point, as the next two examples show.

around the entire circle some number of times. If α is constant, then clearly (0, 0, 0) is not in the image of Ψ. If α is not constant, then for each ζ ∈ ∂D such that α(ζ) ≠ 1, we have β(ζ) = φ∞(α(ζ)): the disc is attached, and that is the only possible z₂-coordinate. In particular, we find a contradiction as this means that β is discontinuous. Therefore, there are no discs attached to X where z₃ = 0 and z₁ is not a unimodular constant. In other words, (0, 0, 0) ∉ DH(X). But (0, 0, 0) is in the closure of DH(X), so DH(X) is not closed.

Example 3.10. The iterated SADH condition is not stable under perturbation. For example, consider M ⊂ ℂ³ given by Im w = |z₁|⁴ - |z₂|⁴. The standard technique of attaching discs one normally uses for a hypersurface with indefinite Levi form applies. These discs fill a neighborhood of the origin, and all shrink to the origin. (Affine linear discs suffice.) However, the perturbation M_ε given by Im w = ε(|z₁|² + |z₂|²) + |z₁|⁴ - |z₂|⁴ is strictly pseudoconvex at 0 for all ε > 0, and hence all analytic discs attached to M_ε near 0 must fall on one side of M_ε. In a similar manner, examples having higher codimension can be constructed.

Example 3.11. Consider the smooth submanifold M ⊂ ℂ³ given in coordinates (z, w₁, w₂) via

w₁ = |z|², w₂ = |z|² + f(|z|²)(Re z),

where f(t) is a smooth real-valued function that is zero precisely when t = 1/n for n ∈ ℕ or t = 0. The submanifold M is of dimension 2. It has a CR singularity at the origin but is totally real at other points. Let φ(ζ) = (z(ζ), w₁(ζ), w₂(ζ)) be an analytic disc attached to M. Then w₁(ζ) and w₂(ζ) are holomorphic functions that are real-valued on ∂D, so they are constant on D̄. This means that |z(ζ)| is also constant on ∂D. If we insist that φ be nonconstant, then z itself must be nonconstant; in particular, Re z must be nonconstant on ∂D. But since |z|² + f(|z|²)(Re z) must be constant on ∂D, we have that f(|z|²) must be zero, which is true only if w₁ = |z|² = 1/n. In other words, the only nonconstant attached analytic discs to M are those in the discrete sequence of discs

φₙ(ζ) = ( 1/√n ζ, 1/n, 1/n ).          (7)

This sequence does "shrink to zero" discretely but not continuously, so it does not give a shrinking disc hull, although these discs are in the regular disc hull DH(M). In particular, these discs cannot be used via the Kontinuitätssatz to extend CR_H functions beyond the initial neighborhood in which they are defined.

Example 3.12. If we modify the preceding example by taking f to be a real-analytic function with finitely many zeros, we find a real-analytic submanifold with trivial topology (topology of a ball) that has only finitely many attached discs.

Example 3.13. Let us show that iteration may be necessary. Consider the set X ⊂ ℂ² given by

X = X₁ ∪ X₂ = { z ∈ ℂ² : |z₁| = |z₂| = 1 and Im z₂ ≥ 0 }
        ∪  { z ∈ ℂ² : |z₁| = 2, |z₂| = 1, and Im z₂ < 0 }.      (8)

If φ = (φ₁, φ₂) : 𝔻 → ℂ² is an analytic disc attached to X, then as the two components of X are disconnected, we have that either φ(∂𝔻) ⊂ X₁ or φ(∂𝔻) ⊂ X₂. Suppose φ(∂𝔻) ⊂ X₁. Then we find that φ₂ must be constant. A similar argument applies if φ(∂𝔻) ⊂ X₂. In either case, the disc then fills in all of |z₁| ≤ 1 or |z₁| ≤ 2. That is, we find that

DH(X) = {z ∈ ℂ² : |z₁| ≤ 1, |z₂| = 1, and Im z₂ ≥ 0}
∪ {z ∈ ℂ² : |z₁| ≤ 2, |z₂| = 1, and Im z₂ ≤ 0}.   (9)

In particular, DH(X) contains the torus 𝕋² given by |z₁| = |z₂| = 1. However, DH(X) does not contain the polydisc 𝔻². It is a relatively routine computation that DH(𝕋²) = 𝔻². Hence DH(X) does not contain the unit polydisc, but DH²(X) does. Thus, iteration is necessary for some sets.

The reader may complain that X is disconnected and that the discs do not all shrink to a point. We modify the previous example as follows. Consider the set X' ⊂ ℂ³ given by

X' = {z ∈ ℂ³ : |z₁| = |z₂| = Re z₃, Im z₂ ≥ 0, z₃ ∈ [0, 1]}
∪ {z ∈ ℂ³ : |z₁| = 2 Re z₃, |z₂| = Re z₃, Im z₂ ≤ 0, z₃ ∈ [0, 1]}.   (10)

By z₃ ∈ [0, 1] we mean that z₃ is real and in the unit interval. The set X' is connected and compact. Since the third component of every analytic disc attached to X' (and hence to any disc hull) must be constant, we reduce the computation to a scaled version of the above example in ℂ². It is not hard to see that similar reasoning holds also for approximately attached discs, and moreover we find that every disc shrinks to the origin. Hence, we have a set X' for which SADH₀(X') = DH(X') and SADH₀(SADH₀(X')) = DH²(X'). In particular, if we wish to extend functions via the hull SADH₀, we must iterate: SADH₀(SADH₀(X')) ≠ SADH₀(X').

Example 3.14. Let us show that a function in CR_H(X) does not necessarily uniquely extend to a holomorphic function in a neighborhood of SADH_q(X) even though it admits analytic continuation to such a neighborhood. Define X ⊂ ℂ³ by

X = {z ∈ ℂ³ : z₁ = tζ, z₂ = 0, z₃ = e^{iπt}, ζ ∈ ∂𝔻, t ∈ [0, 1]}
∪ {z ∈ ℂ³ : z₁ = 0, z₂ = tζ, z₃ = e^{-iπt}, ζ ∈ ∂𝔻, t ∈ [0, 1]}.    (11)

It is clear that the discs ζ ∈ 𝔻 → (tζ, 0, e^{iπt}) and ζ ∈ 𝔻 → (0, tζ, e^{-iπt}) are attached to X, and furthermore these give families shrinking to the point q = (0, 0, 1). It is therefore clear that SADH_q(X) contains the set

{z ∈ ℂ³ : z₁ = tζ, z₂ = 0, z₃ = e^{iπt}, ζ ∈ 𝔻, t ∈ [0, 1]}
∪ {z ∈ ℂ³ : z₁ = 0, z₂ = tζ, z₃ = e^{-iπt}, ζ ∈ 𝔻, t ∈ [0, 1]},  (12)

and any function in CR_H(X) can be analytically continued along the paths given by t ∈ [0, 1] ↦ (0, 0, e^{iπt}) and t ∈ [0, 1] ↦ (0, 0, e^{-iπt}) via the Kontinuitätssatz. The point (0, 0, −1) is in SADH_q(X) but not in X. Take a branch of, say, √z₃ on some neighborhood of X (which we can do since X is simply connected, and so is some thickening of X to obtain a neighborhood). This function is in CR_H(X), but it cannot possibly extend to a CR_H function on SADH_q(X) as the continuation along the two paths given above will give a different value at (0, 0, −1).


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/363135242

Article in Pacific Journal of Mathematics · July 2022

DOI: 10.2140/pjm.2022.319.55

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XIANJING DONG AND SHUANGSHUANG YANG

NEVANLINNA THEORY VIA HOLOMORPHIC FORMS

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This paper redevelops Nevanlinna theory for meromorphic functions on ℂ in the viewpoint of holomorphic forms. According to our observation, Nevanlinna’s functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna’s functions using a holomorphic form 𝒮. With the new settings, an analogue of Nevanlinna theory for the 𝒮-exhausted Riemann surfaces is obtained, which is viewed as a generalization of the classical Nevanlinna theory for ℂ and 𝔻.

Motivation. The paper is motivated by the early work of the first named author [4] (see also Atsuji [1]), who considered Nevanlinna theory for complete Kähler manifolds with nonpositive sectional curvature. To make it simpler, instead of Kähler manifolds, we restrict ourselves to the Riemann surfaces. In what follows, we introduce it without going into the details.

Let S be a noncompact Riemann surface equipped with a complete Hermitian metric h of nonpositive Gauss curvature K. Fix a reference point o ∈ S. Set

κ(r) = min{K(x) : x ∈ D(r)},

where D(r) is a geodesic ball of radius r centered at o. Let f be a nonconstant meromorphic function on S. One can well define Nevanlinna’s functions T_f(r), m_f(r, a) and N_f(r, a) (see Section 3.1 in [4]). The first named author showed that (see Theorem 1.2 in [4]) for any δ > 0

(q - 2)T_f(r) + T(r, ℜ) ≤ ∑_{j=1}^q N_f(r, a_j) + O(log T_f(r) + log C(o, r, δ))

holds for all r ∈ (0, ∞) outside a set E_δ of finite Lebesgue measure, where C(o, r, δ) is a positive function with the estimate (see (19) in [4])

log C(r, o, δ) ≤ O(r√{-κ(r)} + δ log r),

and the curvature term $T(r, \mathcal{R})$ is bounded by

$$r^2 \kappa(r) \leq T(r, \mathcal{R}) \leq 0.$$

As showed above, $\log C(o, r, \delta)$ and $T(r, \mathcal{R})$ are estimated, however, these two estimates are rough and hard to improve by using the previous methods in [4]. For example, we consider $S = \mathbb{D}$, where $\mathbb{D}$ is the unit disc. According to the conditions for metrics, one needs to equip $\mathbb{D}$ with Poincaré metric (of curvature $-1$). In this situation, we have $\kappa(r) \equiv -1$. This gives that

$$-r^2 \leq T(r, \mathcal{R}) \leq 0.$$

However, the best lower bound of $T(r, \mathcal{R})$ is $O(-r)$.

The main drawback of the method in [4] is that the selectivity of metrics is restricted, i.e., the metrics have to be complete and nonpositively curved, but this will result in a rough estimate.

In this paper, we shall investigate Nevanlinna theory from the viewpoint of holomorphic forms $\mathcal{S}$. With this idea, one generalizes the Nevanlinna theory to one class of noncompact Riemann surfaces which we call the $\mathcal{S}$-exhausted Riemann surfaces. To begin with, let us describe how Nevanlinna's functions can be formulated by a holomorphic form $\mathcal{S}$. To see it more clearly, we shall use the Ahlfors' characteristic function

$$(1) \qquad T_f(r) = \int_1^r \frac{dt}{t} \int_{D(t)} f^* \omega_{FS},$$

A viewpoint of holomorphic forms. Let $f$ be a nonconstant meromorphic function on $\mathbb{C}$, namely, the complex plane with standard Euclidean metric. Nevanlinna's characteristic function $T(r, f)$ of $f$ is well known [10] as

$$T(r, f) := m(r, f) + N(r, f),$$

where

$$m(r, f) = \int_{|z|=r} \log^+ |f| \frac{d\theta}{2\pi}, \quad N(r, f) = \int_0^r \frac{n(t, f) - n(0, f)}{t} dt + n(0, f) \log r$$

are called the proximity function and counting function of $f$ respectively, in which, $n(r, f)$ denotes the number of poles of $f$ on the disc $D(r) := \{|z| < r\}$. Characteristic function is an important notion, it characterizes the growth of meromorphic functions, using which Nevanlinna [10] in 1925 established two fundamental theorems: First Main Theorem and Second Main Theorem, named Nevanlinna theory. Nevanlinna theory plays a central role in complex analysis and hyperbolic geometry. Roughly speaking, this theory studies the size of images of meromorphic functions or mappings in target spaces; it is a great generalization of the Little Picard Theorem saying that a meromorphic function must be a constant if it omits three distinct values in $\mathbb{C}$.

which is equivalent to T(r, f) up to a bounded term, here ω_FS is the Fubini–Study form on ℙ¹(ℂ). We need the proximity function

(2)        mf(r, a) = ∫_{|z|=r} log (1/∥f, a∥) (dθ/2π),

where ∥·,·∥ is the spherical distance on ℙ¹(ℂ). Taking the holomorphic form 𝒮 = dz, we will see that ℂ is 𝒮-exhausted (see Definition 2.6). Define

ẑ = ∫₀^z 𝒮,

which is a holomorphic function in z with a unique zero z = 0. The 𝒮-disc and 𝒮-circle of radius r centered at 0 are defined respectively by

D^𝒮(r) = {z : |ẑ| < r} and C^𝒮(r) = {z : |ẑ| = r}.

Let g_r(0, z) be the Green function of Δ/2 for D^𝒮(r) with a pole at 0 and Dirichlet boundary condition and let dπ₀^r be the harmonic measure on C^𝒮(r) with respect to 0. Notice that g_r(0, z) = (1/π) log(r/|ẑ|), dπ₀^r = dθ/2π, then by integration by parts, (1) can be rewritten as

Tf(r) = ∫₁^r (dt/t) ∫_{D^𝒮(t)} f^*ω_FS = π ∫_{D^𝒮(r)} g_r(0, z) f^*ω_FS − π ∫_{D^𝒮(1)} g₁(0, z) f^*ω_FS

and (2) can be replaced by

mf(r, a) = ∫_{C^𝒮(r)} log (1/∥f, a∥) dπ₀^r.

Similarly, the counting function is that

Nf(r, a) = ∫₁^r (n_f^𝒮(t, a)/t) dt,

where n_f^𝒮(r, a) denotes the number of zeros of f − a on D^𝒮(r).

Follow the idea as shown above, we consider a noncompact Hermitian Riemann surface (𝒮, h). Choosing 𝒮, a nowhere-vanishing holomorphic form on 𝒮 such that ẋ = ∫ₒˣ 𝒮 defines a holomorphic function, where o is a fixed point in 𝒮. Following [6], such a form always exists. If 𝒮 is 𝒮-exhausted, i.e., any sequence of 𝒮-discs exhausts 𝒮 when radius approaches increasingly to R^𝒮, where R^𝒮 is the 𝒮-radius of 𝒮 with respect to o, see Definition 2.7. To a 𝒮-exhausted surface 𝒮, T_f^𝒮(r), m_f^𝒮(r, a) and N_f^𝒮(r, a) of a meromorphic function f on 𝒮 can be similarly defined and they make sense, see Notations on page 63 for the definition. By computing Green functions and harmonic measures, we shall establish an analogue of Nevanlinna theory, which turns out to be an extension of the classical Nevanlinna theory for ℂ and 𝔻 (unit disc); see, e.g., [10; 11; 14; 15].

Main results. In what follows, we state the main results of the paper.

Theorem I (Theorem 3.4). Let \((S, h; \mathscr{S})\) be a \(\mathscr{S}\)-exhausted Hermitian Riemann surface of \(\mathscr{S}\)-radius \(R^{\mathscr{S}}\) with respect to \(o\). Let \(\gamma\) be an integrable function on \((0, R^{\mathscr{S}})\) with \(\int_0^{R^{\mathscr{S}}} \gamma(r) dr = \infty\). Let \(f\) be a nonconstant meromorphic function on \(S\) and \(a_1, \ldots, a_q\) be distinct values in \(\mathbb{C}\). Then for any \(\delta > 0\)
\[
(q - 2) T_f^{\mathscr{S}}(r) + T^{\mathscr{S}}(r, \mathscr{R}) \leq \sum_{j=1}^q \overline{N}_f^{\mathscr{S}}(r, a_j) + O(\log T_f^{\mathscr{S}}(r) + \log \|\mathscr{S}\|_{r,\sup} + \log \gamma(r) + \delta \log r)
\]
holds for all \(r \in (0, R^{\mathscr{S}})\) outside a set \(E_\delta\) with \(\int_{E_\delta} \gamma(r) dr < \infty\), where
\[
\|\mathscr{S}\|_{r,\sup} = \sup\{ \|\mathscr{S}_x\|_h : |\hat{x}| < r \}.
\]

In Theorem I, the term \(T^{\mathscr{S}}(r, \mathscr{R})\) is called the characteristic of the Ricci form \(\mathscr{R} := -dd^c \log h\) of \(S\), which depends on the Gauss curvature of \(S\), see (8). Notice that \(\mathbb{C}\) is \(dz\)-exhausted and \(\|dz\| = 1\) under standard Euclidean metric. In this case, the Nevanlinna's functions agree with the classical ones. Hence, Theorem I yields (by letting \(\gamma = 1\)) a classical consequence for \(\mathbb{C}\) that:

Corollary I. Let \(f\) be a nonconstant meromorphic function on \(\mathbb{C}\), and let \(a_1, \ldots, a_q\) be distinct points in \(\mathbb{C}\). Then for any \(\delta > 0\)
\[
(q-2) T_f(r) \leq \sum_{i=1}^q \overline{N}_f(r, a_j) + O(\log T_f(r) + \delta \log r)
\]
holds for all \(r \in (0, \infty)\) outside a set \(E_\delta\) of finite Lebesgue measure.

Equipping \(\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}\) with standard Euclidean metric, then \(\mathbb{D}\) is a \(dz\)-exhausted Riemann surface. Take \(\gamma = (1-r)^{-1}\), it concludes another classical consequence for \(\mathbb{D}\) that:

Corollary II. Let \(f\) be a nonconstant meromorphic function on \(\mathbb{D}\), and let \(a_1, \ldots, a_q\) be distinct points in \(\mathbb{C}\). Then for any \(\delta > 0\)
\[
(q-2) T_f(r) \leq \sum_{i=1}^q \overline{N}_f(r, a_j) + O\left( \log T_f(r) + \log \frac{1}{1-r} \right)
\]
holds for all \(r \in (0, 1)\) outside a set \(E_\delta\) with \(\int_{E_\delta} (1-r)^{-1} dr < \infty\).

Associating a holomorphic mapping \(f : S \to \mathscr{R}\), where \((\mathscr{R}, \omega)\) is a compact Hermitian Riemann surface. Following a theorem of Chern [3], we can similarly define Nevanlinna's functions via \(\mathscr{S}\). We prove a more generalized theorem of Theorem I:

Theorem II (Theorem 4.1). Let \((S, h; \mathscr{S})\) be a \(\mathscr{S}\)-exhausted Hermitian Riemann surface of \(\mathscr{S}\)-radius \(R^{\mathscr{S}}\) with respect to \(o\), and let \(\mathscr{R}\) be a compact Riemann surface of genus \(g\). Fix a positive \((1,1)\)-form \(\omega\) on \(\mathscr{R}\). Let \(\gamma\) be an integrable function on \((0, R^{\mathscr{S}})\) with \(\int_0^{R^{\mathscr{S}}} \gamma(r)\, dr = \infty\). Let \(f\) be a nonconstant holomorphic mapping from \(S\) into \(\mathscr{R}\) and \(a_1, \ldots, a_q\) be distinct points in \(\mathscr{R}\). Then for any \(\delta > 0\)

\[
(q - 2 + 2g)T_{f,\omega}^{\mathscr{S}}(r) + T^{\mathscr{S}}(r, \mathscr{R})
\leq \sum_{j=1}^q \overline{N}_f^{\mathscr{S}}(r, a_j) + O\big(\log T_{f,\omega}^{\mathscr{S}}(r) + \log \|\mathscr{S}\|_{r,\sup} + \log \gamma(r) + \delta \log r\big)
\]

holds for all \(r \in (0, R^{\mathscr{S}})\) outside a set \(E_\delta\) with \(\int_{E_\delta} \gamma(r) dr < \infty\), where

\[
\|\mathscr{S}\|_{r,\sup} = \sup\{\|\mathscr{S}_x\|_h : |\hat{x}| < r\}.
\]

Remark 1.1. In order to better estimate \(\|\mathscr{S}\|\), we can put Hermitian metric \(\alpha = \frac{\sqrt{-1}}{2} \mathscr{S} \wedge \overline{\mathscr{S}}\) induced by \(\mathscr{S}\) on \(S\). Under this metric, we have \(\|\mathscr{S}\| \equiv 1\). Then the result in Theorem II becomes

\[
(q - 2 + 2g)T_{f,\omega}^{\mathscr{S}}(r) + T^{\mathscr{S}}(r, \mathscr{R})
\leq \sum_{j=1}^q \overline{N}_f^{\mathscr{S}}(r, a_j) + O(\log T_{f,\omega}^{\mathscr{S}}(r) + \log \gamma(r) + \delta \log r).
\]

We consider a defect relation of \(f\) in Nevanlinna theory. The simple defect \(\overline{\delta}_f(a)\) of \(f\) with respect to \(a\) is defined by

\[
(3) \qquad \overline{\delta}_f(a) = 1 - \limsup_{r \to R^{\mathscr{S}}} \frac{\overline{N}_f^{\mathscr{S}}(r, a)}{T_{f,\omega}^{\mathscr{S}}(r)}.
\]

If \(f\) is nonconstant, then we can check \(T_{f,\omega}^{\mathscr{S}}(r) \geq O(\log r)\). By using the First Main Theorem (see page 65), we see that \(0 \leq \overline{\delta}_f(a) \leq 1\).

By estimating the lower bound of \(T^{\mathscr{S}}(r, \mathscr{R})\), we obtain a defect relation:

Theorem III. Assume the same conditions as in Theorem II. Suppose, in addition, that \(-C \leq K \leq 0\) for a nonnegative constant \(C\). If \(f\) satisfies

\[
\limsup_{r \to R^{\mathscr{S}}} \frac{Cr^2 \|\mathscr{S}\|_{r,\inf}^{-1} + \log(\gamma(r)\|\mathscr{S}\|_{r,\sup})}
{T_{f,\omega}^{\mathscr{S}}(r)} = 0,
\]

where

\[
\|\mathscr{S}\|_{r,\inf} = \inf\{\|\mathscr{S}_x\|_h : |\hat{x}| < r\},
\]

then we have the defect relation

\[
\sum_{j=1}^q \overline{\delta}_f(a_j) \leq 2 - 2g.
\]

The earlier study of Nevanlinna theory for noncompact Riemann surfaces dates back to the work of H. Wu [17] (also refer to Shabat [16]). More details on this aspect, the readers may refer to the recent very nice papers of He and Ru [7] and Păun and Sibony [13]. We have to indicate that the paper presents a new method of studying Nevanlinna theory by putting a metric \( h \) on \( S \). However, we should also point out that our results can be derived by Wu's method (as well as He and Ru's arguments) since \( \mathscr{S} \) produces an exhaustion function

\[
\sigma(x) := \left| \int_o^x \mathscr{S} \right|
\]

on \( S \), so that \( \log \sigma^2 \) is harmonic outside a compact set. We refer the readers to the standard arguments of He and Ru [7] without going into any details.

\emph{\(\mathscr{S}\)-exhausted Riemann surfaces.} Let \( S \) be a noncompact Riemann surface. Gunning proved that:

\textbf{Proposition 2.1} (Gunning and Narasimhan [6]). Let \( S \) be a noncompact Riemann surface. Then there is a holomorphic form \( \mathscr{S} \) on \( S \) satisfying the conditions:

(a) \( \mathscr{S} \) has no zeros on \( S \);
(b) \( \int_\gamma \mathscr{S} = 0 \) for each smooth simple closed curve \( \gamma \) in \( S \).

\textbf{Remark 2.2.} Condition (a) defines a holomorphic field \( X \) without zeros on \( S \) in the following manner: write \( \mathscr{S} = \phi\,dz \) in a local holomorphic coordinate \( z \), it is trivial to check that

\[
X = \phi^{-1} \frac{\partial}{\partial z}
\]

is well defined on \( S \). Hence, \( X \) is a nowhere-vanishing holomorphic field that is dual to \( \mathscr{S} \). Condition (b) defines a holomorphic function \( \hat{x} : S \to \mathbb{C} \) by

\[
(4) \qquad L(x) = \int_o^x \mathscr{S} := \hat{x},
\]

where \( o \) is a fixed point in \( S \). By (a), we see that \( o \) is a simple zero of \( L \).

In this paper, besides conditions (a) and (b), one assumes that \( \mathscr{S} \) satisfies an additional condition: \( L(x) = 0 \) if and only if \( x = o \). This condition forces \( L \) to be a univalent function.

\textbf{Definition 2.3.} Let \( \mathscr{S} \) be a holomorphic form on \( S \) satisfying the conditions (a) and (b) in Proposition 2.1. If, in addition, that \( \mathscr{S} \) satisfies the condition: \( L(x) = 0 \) if and only if \( x = o \), where \( L \) is defined by (4), then we say that \( S \) is a \( \mathscr{S} \)-Riemann surface.

We define \( \mathscr{S} \)-discs and \( \mathscr{S} \)-circles on a \( \mathscr{S} \)-Riemann surface \( S \).

Definition 2.4. Let $(S; \mathscr{S})$ be a $\mathscr{S}$-Riemann surface. The $\mathscr{S}$-disc $D^{\mathscr{S}}(r)$ is defined by

$$D^{\mathscr{S}}(r) = \{x \in S : |\hat{x}| < r\},$$

and the $\mathscr{S}$-circle $C^{\mathscr{S}}(r)$ is defined by

$$C^{\mathscr{S}}(r) = \{x \in S : |\hat{x}| = r\}.$$

Since $L$ is univalent, then $D^{\mathscr{S}}(r)$ is simply connected for all $r > 0$. Notice that the case $C^{\mathscr{S}}(r) = \emptyset$ may happen if $r$ is sufficiently large, since $D^{\mathscr{S}}(r)$ could cover the whole surface $S$ when $r$ is large enough in some situations. For example, we consider the case where $S = \mathbb{D}$ and $\mathscr{S} = dz$, then $C^{\mathscr{S}}(r) = \{z \in \mathbb{D} : |z| = r\}$. If $r > 1$, then we have $C^{\mathscr{S}}(r) = \emptyset$.

Definition 2.5. We say that $\mathscr{S}$ is an exhaustion form, if $\{D^{\mathscr{S}}(r_n)\}_{n=1}^{\infty}$ exhausts $S$ whenever $0 < r_1 < r_2 < \cdots$ and $r_n \to +\infty$ as $n \to +\infty$. Namely, $S = \bigcup_{n=1}^{\infty} D^{\mathscr{S}}(r_n)$ and one of the following holds:

(i) $D^{\mathscr{S}}(r_1) \subset \overline{D^{\mathscr{S}}(r_1)} \subset D^{\mathscr{S}}(r_2) \subset \overline{D^{\mathscr{S}}(r_2)} \subset \cdots \subset S$.

(ii) $D^{\mathscr{S}}(r_1) \subset \overline{D^{\mathscr{S}}(r_1)} \subset \cdots \subset D^{\mathscr{S}}(r_k) = D^{\mathscr{S}}(r_{k+1}) = \cdots = S$.

Precisely, we say that $\mathscr{S}$ is a parabolic exhaustion form if condition (i) is satisfied, and a hyperbolic exhaustion form if condition (ii) is satisfied.

Definition 2.6. Let $(S; \mathscr{S})$ be a $\mathscr{S}$-Riemann surface. We say that $S$ is a $\mathscr{S}$-exhausted Riemann surface if $\mathscr{S}$ is an exhaustion form on $S$.

If $S$ is a $\mathscr{S}$-exhausted Riemann surface, then $C^{\mathscr{S}}(r)$ (for $r > 0$) is a closed curve contained properly in $S$ whenever $C^{\mathscr{S}}(r) \cap S \neq \emptyset$.

Definition 2.7. Let $(S; \mathscr{S})$ be a $\mathscr{S}$-exhausted Riemann surface. Define $R^{\mathscr{S}}$ by

$$R^{\mathscr{S}} = \sup\{r > 0 : C^{\mathscr{S}}(r) \cap S \neq \emptyset\},$$

which is called the $\mathscr{S}$-radius of $S$ with respect to $o$.

From the definition as above, we have that $0 < R^{\mathscr{S}} \leq \infty$ and $C^{\mathscr{S}}(r)$ is a simple closed curve in $S$ for $0 < r < R^{\mathscr{S}}$. Note that the following corollary is a direct consequence of Definitions 2.6 and 2.7.

Corollary 2.8. Let $(S; \mathscr{S})$ be a $\mathscr{S}$-exhausted Riemann surface of $\mathscr{S}$-radius $R^{\mathscr{S}}$ with respect to $o$. Let $\{r_n\}_{n=1}^{\infty}$ be a sequence of positive integers $r_n$ with $r_1 < r_2 < \cdots$ and $r_n \to R^{\mathscr{S}}$ as $n \to +\infty$. Then $\{D^{\mathscr{S}}(r_n)\}_{n=1}^{\infty}$ exhausts $S$.

Example 2.9. Let $\mathbb{C}$ be the standard complex Euclidean plane, then $\mathbb{C}$ is a $dz$-exhausted Riemann surface of $dz$-radius $\infty$ with respect to $0$. We obtain $\hat{z} = z$, $D^{dz}(r) = \{z \in \mathbb{C} : |z| < r\}$ and $C^{dz}(r) = \{z \in \mathbb{C} : |z| = r\}$.

Example 2.10. Equipping 𝔻 with standard Euclidean metric, then 𝔻 is a dz-exhausted Riemann surface of dz-radius 1 with respect to 0. We have ẑ = z, D^{dz}(r) = {z ∈ 𝔻 : |z| < r} and C^{dz}(r) = {z ∈ 𝔻 : |z| = r}.

Harmonic measures on 𝒮-circles. Let (𝒮, h; 𝒮) be a 𝒮-exhausted Hermitian Riemann surface of 𝒮-radius R^{𝒮} with respect to o. We denote by Δ the Laplace-Beltrami operator on 𝒮 defined by h. For 0 < r < R^{𝒮}, we shall compute Green function g_r(o, x) of ½Δ for D^{𝒮}(r) with Dirichlet boundary and a pole at o, i.e.,

  -½Δ g_r(o, x) = δ_o(x), x ∈ D^{𝒮}(r); g_r(o, x) = 0, x ∈ C^{𝒮}(r)

in the sense of distributions, as well as harmonic measure dπ_o^r on C^{𝒮}(r) with respect to o, for 0 < r < R^{𝒮}. We have the well-known formula

  dπ_o^r(x) = -½ ∂g_r(o, x)/∂n dσ_r(x),

where ∂/∂n is the inward normal derivative on C^{𝒮}(r).

Lemma 2.11. For 0 < r < R^{𝒮}, we have

  g_r(o, x) = 1/π log(r/|ẑ|).

Proof. Since L is a holomorphic function on 𝒮 with a unique simple zero o, then it follows that

  (1/2π) Δ log |L(x)| = δ_o(x), x ∈ D^{𝒮}(r)

in the sense of distribution. Indeed, it is clear that |L(x)| = r for x ∈ C^{𝒮}(r). Hence, we obtain

  g_r(o, x) = (1/π) log(r/|ẑ|).

□

Lemma 2.12. For 0 < r < R^{𝒮}, we have

  dπ_o^r = (‖𝒮‖ₕ)/(2πr) dσ_r,

where dσ_r is the Riemannian line element on C^{𝒮}(r) defined by h.

Proof. Fix an arbitrary point x₀ ∈ C^{𝒮}(r), we take a holomorphic coordinate z near x₀ such that h|_{x₀} = 1. Note that

  dπ_o^r(x) = -½ ∂g_r(o, x)/∂n dσ_r(x),

where ∂/∂n is the inward normal derivative on ∂C^{𝒮}(r). By |L(x₀)| = r, we have

  dπ_o^r(x₀) = (1/2πr) ∂|L(x₀)|/∂n dσ_r(x₀).

Set \( z = \xi_1 + \sqrt{-1}\xi_2 \). Differentiating the equation \( |L| = r \) near \( x_0 \), then
\[
\frac{\partial |L|}{\partial \xi_1} d\xi_1 + \frac{\partial |L|}{\partial \xi_2} d\xi_2 = 0,
\]
which gives an inward normal vector
\[
\vec{n}|_{x_0} = \left( \frac{\frac{\partial |L(x_0)|}{\partial \xi_1}}{\sqrt{ \left( \frac{\partial |L(x_0)|}{\partial \xi_1} \right)^2 + \left( \frac{\partial |L(x_0)|}{\partial \xi_2} \right)^2 }}, \frac{\frac{\partial |L(x_0)|}{\partial \xi_2}}{\sqrt{ \left( \frac{\partial |L(x_0)|}{\partial \xi_1} \right)^2 + \left( \frac{\partial |L(x_0)|}{\partial \xi_2} \right)^2 }} \right).
\]

Write \( \mathscr{S} = \phi \, dz \) in the local coordinate \( z \), a direct computation follows that
\[
\frac{\partial |L(x_0)|}{\partial \xi_1} = \Re \left[ \phi(x_0) \frac{\overline{L(x_0)}^{1/2}}{L(x_0)^{1/2}} \right], \quad
\frac{\partial |L(x_0)|}{\partial \xi_2} = -\Im \left[ \phi(x_0) \frac{\overline{L(x_0)}^{1/2}}{L(x_0)^{1/2}} \right].
\]
Then
\[
\frac{\partial |L(x_0)|}{\partial \vec{n}} = \left( \frac{\partial |L(x_0)|}{\partial \xi_1}, \frac{\partial |L(x_0)|}{\partial \xi_2} \right) \cdot \vec{n}|_{x_0} = |\phi(x_0)| = \| \mathscr{S}x_0 \|_h.
\]

We conclude that
\[
d\pi_o^r(x_0) = \frac{\| \mathscr{S}x_0 \|_h}{2\pi r}\, d\sigma_r(x_0). \quad \Box
\]

Notations. Let \( (\mathscr{S}, h; \mathscr{S}) \) be a \( \mathscr{S} \)-exhausted Hermitian Riemann surface of \( \mathscr{S} \)-radius \( R^{\mathscr{S}} \) with respect to \( o \). Using \( \mathscr{S} \), we extend the notion of Nevanlinna’s functions to \( \mathscr{S} \). Let \( f \) be a meromorphic function on \( \mathscr{S} \) and let \( r \) be satisfied with \( 0 < r_0 < r < R^{\mathscr{S}} \). Viewing \( f = f_1/f_0 = [f_0 : f_1] \) as a holomorphic mapping from \( \mathscr{S} \) into \( \mathbb{P}^1(\mathbb{C}) \). We define the characteristic function of \( f \) by

\[
T_f^{\mathscr{S}} (r) = \int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} f^* \omega_{FS},
\]

where \( \omega_{FS} = dd^c \log \left( |\zeta_0|^2 + |\zeta_1|^2 \right) \) and

\[
d = \partial + \bar{\partial}, \quad d^c = \frac{\sqrt{-1}}{4\pi} (\bar{\partial} - \partial), \quad dd^c = \frac{\sqrt{-1}}{2\pi} \partial \bar{\partial}.
\]

By integration by part, it yields that

\[
T_f^{\mathscr{S}} (r) = \int_{D^{\mathscr{S}}(r)} \log \frac{r}{|\hat{x}|} \, dd^c \log \left( |f_0(x)|^2 + |f_1(x)|^2 \right) - \int_{D^{\mathscr{S}}(r_0)} \log \frac{r_0}{|\hat{x}|} \, dd^c \log \left( |f_0(x)|^2 + |f_1(x)|^2 \right).
\]

Locally, we write

\[
dV = \sqrt{-1} h \, dz \wedge d\bar{z}, \quad \Delta = \frac{2}{h} \frac{\partial^2}{\partial z \partial \bar{z}},
\]

where \( dV \) is the Riemannian area element of \( S \). We get

\[
dd^c \log (|f_0|^2 + |f_1|^2) = \frac{1}{4\pi} \Delta \log (|f_0|^2 + |f_1|^2) dV.
\]

Notice (see Lemma 2.11) that

\[
g_r(o, x) = \frac{1}{\pi} \log \frac{r}{|\hat{x}|},
\]

then \( T_f^S(r) \) can be written in terms of Green function as

\[
(5)\quad T_f^S(r) = \frac{1}{4} \int_{D^S(r)} g_r(o, x) \Delta \log (|f_0(x)|^2 + |f_1(x)|^2) dV(x)
- \frac{1}{4} \int_{D^S(r_0)} g_{r_0}(o, x) \Delta \log (|f_0(x)|^2 + |f_1(x)|^2) dV(x).
\]

Let \( a = [a_0 : a_1] \in \mathbb{P}^1(\mathbb{C}) \) such that \( f \not\equiv a \). The proximity function of \( f \) with respect to \( a \) is defined by

\[
(6)\quad m_f^S(r, a) = \int_{C^S(r)} \log \frac{1}{\|f, a\|} d\pi_o^r,
\]

where \( \|\cdot, \cdot\| \) is the spherical distance on \( \mathbb{P}^1(\mathbb{C}) \), defined by

\[
\|f, a\| = \frac{|\langle f ; a \rangle|}{\|f\|\|a\|} = \frac{|a_0 f_1 - a_1 f_0|}{\sqrt{|a_0|^2 + |a_1|^2} \sqrt{|f_0|^2 + |f_1|^2}},
\]

where

\[
\langle f ; a \rangle := a_0 f_1 - a_1 f_0.
\]

Using Lemma 2.11, we obtain

\[
m_f^S(r, a) = \frac{1}{2\pi r} \int_{C^S(r)} \|S\|_h \log \frac{1}{\|f, a\|} d\sigma_r,
\]

where \( d\sigma_r \) is the Riemannian line element on \( C^S(r) \). The counting function of \( f \) with respect to \( a \) is defined by

\[
N_f^S(r, a) = \int_{r_0}^r \frac{n_f^S(t, a)}{t} dt,
\]

where \( n_f^S(r, a) \) denotes the number of the zeros of \( f - a \) on \( D^S(r) \) counting multiplicities. By Poincaré–Lelong formula [2], we see that

\[
N_f^S(r, a) = \int_{r_0}^r \frac{dt}{t} \int_{D^S(t)} dd^c [\log |\langle f ; a \rangle|^2],
\]

where \( dd^c [\log |\langle f ; a \rangle|^2] \) is a current [11]. Similarly, in terms of Green function, there is an alternate expression:

\[
(7)\quad N_f^S(r, a) = \frac{1}{4} \int_{D^S(r)} g_r(o, x) \Delta \log |\langle f(x); a \rangle|^2 dV(x)
- \frac{1}{4} \int_{D^S(r_0)} g_{r_0}(o, x) \Delta \log |\langle f(x); a \rangle|^2 dV(x),
\]

where Δ log |⟨f ; a⟩|² is understood as a distribution. Similarly, one can define the simple counting function N̄ᶠˢ(r, a) of f with respect to a, which measures the size of the set of zeros of f − a without counting multiplicities.

We define several other symbols. Let ℛ be the Ricci form of (S, h), i.e.,
$$\mathcal{R} = \operatorname{Ric}(\alpha) = -dd^{c} \log h,$$
where
$$\alpha = \frac{\sqrt{-1}}{\pi} h\,dz \wedge d\bar{z}$$
is the Kähler form of S. The characteristic of ℛ is defined by
$$T^{\mathscr{S}}(r,\mathcal{R}) = \int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} \mathcal{R}.$$
Since the Gauss curvature K of h is computed by
$$K = -\frac{1}{2}\Delta \log h = -\frac{1}{h} \frac{\partial^2 \log h}{\partial z \partial \bar{z}},$$
then we obtain
$$2\mathcal{R} = K \alpha.$$
By this with the above, we see that the characteristic of ℛ can be expressed in terms of Gauss curvature that
$$(8) \qquad T^{\mathscr{S}}(r,\mathcal{R}) = \frac{1}{2} \int_{D^{\mathscr{S}}(r)} g_{r}(o,x)K(x)dV(x) - \frac{1}{2} \int_{D^{\mathscr{S}}(r_0)} g_{r_0}(o,x)K(x)dV(x).$$

First Main Theorem. Let us introduce Dynkin formula which is a generalization of Green–Jensen formula [12; 14], see the probabilistic version of Dynkin formula in [4; 8; 9].

Lemma 3.1 (Dynkin formula). Let u be a function of 𝒞²-class except at most a polar set of singularities on a Riemannian manifold M. Let D ⊂ M be a relatively compact domain with piecewise smooth boundary ∂D. Assume that u(o) ≠ ∞ for a fixed point o ∈ D. Then
$$\int_{\partial D} u(x) d\pi_o^{\partial D}(x) - u(o) = \frac{1}{2} \int_D g_D(o,x)\Delta u(x) dV(x),$$
where g_D(o,x) is the Green function of Laplacian Δ/2 for D with a pole o and Dirichlet boundary condition, and dπ_o^{∂D} is the harmonic measure on ∂D with respect to o. Here, Δu is understood as a distribution.

Proof. We are to prove the above lemma by using a probabilistic approach, i.e., the probabilistic Dynkin formula is applied to showing this lemma. Let X_t := {X_t}_{t\geq0} be the Brownian motion generated by Δ/2, started at o ∈ M (see Section 2.2 in [4]).

Denote by \(\mathbb{P}_o\) the law or distribution of \(X_t\) starting from \(o\) and by \(\mathbb{E}_o\) the expectation of \(X_t\) with respect to \(\mathbb{P}_o\). Set the stopping time

\[
\tau_D = \inf\{t > 0 : X_t \notin D\}.
\]

Note that the probabilistic Dynkin formula (see Itô formula in [4], page 6) says that

\[
\mathbb{E}_o[u(X_{\tau_D})] - u(o) = \frac{1}{2} \mathbb{E}_o\left[\int_0^{\tau_D} \Delta u(X_t)\, dt\right].
\]

On the other hand, using the co-area formula (see (2) in [4]) and the relation between harmonic measures and hitting times (see (3) in [4]), we obtain

\[
\mathbb{E}_o\left[\int_0^{\tau_D} \Delta u(X_t)\, dt\right] = \int_D g_D(o, x)\Delta u(x)\, dV(x)
\]
and
\[
\mathbb{E}_o[u(X_{\tau_D})] = \int_{\partial D} u(x)\, d\pi_o^{\partial D}(x).
\]

Substituting the two equalities into the above probabilistic Dynkin formula, we have the lemma proved. \(\Box\)

Applying Dynkin formula to \(\log \|f, a\|^{-1}\) and noting (5)–(7), it follows the First Main Theorem.

## Theorem 3.2 (First Main Theorem).
If \(f \not\equiv a\), then we have
\[
T_f^{\mathscr{S}}(r) = m_f^{\mathscr{S}}(r, a) + N_f^{\mathscr{S}}(r, a) + O(1).
\]

### Remark 3.3.
Theorem 3.2 can be confirmed by using Green–Jensen formula instead of Dynkin formula, since \(\mathscr{S}\) can produce a parabolic or a hyperbolic exhaustion function \(\sigma(x) = |\int_o^x \mathscr{S}|\); see, e.g., [7; 13; 16; 17].

## Second Main Theorem.
The main purpose here is to prove the following theorem.

## Theorem 3.4 (Second Main Theorem).
Let \((S, h; \mathscr{S})\) be a \(\mathscr{S}\)-exhausted Hermitian Riemann surface of \(\mathscr{S}\)-radius \(R^{\mathscr{S}}\) with respect to \(o\). Let \(\gamma\) be an integrable function on \((0, R^{\mathscr{S}})\) with \(\int_0^{R^{\mathscr{S}}} \gamma(r)\, dr = \infty\). Let \(f\) be a nonconstant meromorphic function on \(S\) and \(a_1, \ldots, a_q\) be distinct values in \(\mathbb{C}\). Then for any \(\delta > 0\)

\[
(q - 2)T_f^{\mathscr{S}}(r) + T^{\mathscr{S}}(r, \mathscr{R})
\leq \sum_{j=1}^q N_f^{\mathscr{S}}(r, a_j) + O\left(\log T_f^{\mathscr{S}}(r) + \log \|\mathscr{S}\|_{r,\sup} + \log \gamma(r) + \delta \log r\right)
\]

holds for all \(r \in (0, R^{\mathscr{S}})\) outside a set \(E_\delta\) with \(\int_{E_\delta} \gamma(r) dr < \infty\), where
\[
\|\mathscr{S}\|_{r,\sup} = \sup\{\|\mathscr{S}_x\|_h : |\hat{x}| < r\}.
\]

The following is called the Borel's Growth Lemma.

We utilize Borel’s growth lemma to prove the following calculus lemma. Let \( k \) be a locally integrable function on \( S \). Set

\[
E_k(r) = \int_{C^S(r)} k\,d\tau_o^r, \quad A_k(r) = \int_{r_0}^r \frac{dt}{t} \int_{D^S(t)} k\,dV.
\]

**Lemma 3.6.** Let \((S, h; S)\) be a \(S\)-exhausted Hermitian Riemann surface of \(S\)-radius \(R^S\) with respect to \(o\). Let \(\gamma\) be an integrable function on \((0, R^S)\) with \(\int_0^{R^S} \gamma(r)\,dr = \infty\). Let \(k\) be a locally integrable function on \(S\). Then for any \(\delta > 0\)
\[
E_k(r) \leq \frac{\|S\|_{r, \sup} r^{\delta} \gamma^{2+\delta}(r)}{2\pi} A_k^{(1+\delta)^2}(r)
\]
holds for all \( r \in (0, R^S) \) outside a set \( E_\delta \) with \(\int_{E_\delta} \gamma(r)\, dr < \infty\), where
\[
\|S\|_{r, \sup} = \sup\{\|S_x\|_h : |\hat{x}| < r \}.
\]

*Proof.* Notice that
\[
\int_{D^S(r)} k\,dV = \int_0^r dt \int_{C^S(t)} k\,d\sigma_r,
\]
then it follows from Lemma 2.12 that
\[
\frac{d}{dr}\left(r \frac{dA_k}{dr}\right) = \int_{C^S(r)} k\,d\sigma_r \geq \frac{2\pi r}{\|S\|_{r,\sup}} E_k(r).
\]
Using Lemma 3.5 twice (first to \(rA_k'\) and then to \(A_k\)), we obtain
\[
E_k(r) \leq \frac{\|S\|_{r,\sup} r^\delta \gamma^{2+\delta}(r)}{2\pi} A_k^{(1+\delta)^2}(r). \quad \square
\]

**Lemma 3.5** [15]. Let \(\gamma\) be an integrable function on \((0, R)\) with \(\int_0^R \gamma(r) dr = \infty\).
Let \(h\) be a nondecreasing function of \(C^1\)-class on \((0, R)\). Let \(\lim_{r \to R} h(r) = \infty\) and \(h(r_0) > 0\) for some \(r_0 \in (0, R)\). Then for any \(\delta > 0\)
\[
h'(r) \leq h^{1+\delta}(r) \gamma(r)
\]
holds for all \(r \in (0, R)\) outside a set \(E_\delta\) with \(\int_{E_\delta} \gamma(r) dr < \infty\). In particular, when \(R = \infty\), we can take \(\gamma = 1\). Then for any \(\delta > 0\)
\[
h'(r) \leq h^{1+\delta}(r)
\]
holds for all \(r \in (0, \infty)\) outside a set \(E_\delta\) of finite Lebesgue measure.

**Proof of Theorem 3.4.** Consider a singular volume form (see [2; 5])
\[
(9) \quad \Phi = \frac{C\omega_{FS}}{\prod_{j=1}^q \|\zeta, a_j\|^2 \log^2 \|\zeta, a_j\|^{-2}}
\]

on \(\overline{\mathbb{C}}\), where \(a_1, \ldots, a_q\) are distinct values in \(\overline{\mathbb{C}}\) and \(\omega_{FS} = dd^c \log(1 + |\zeta|^2)\). Since \(\overline{\mathbb{C}}\) is compact, we can choose a positive number \(C\) such that \(\int_{\overline{\mathbb{C}}} \Phi = 1\). Set

\[
f^* \Phi = \xi \frac{\sqrt{-1}}{\pi} h\, dz \wedge d\bar{z}.
\]

By taking the Ricci form of both sides of the above identity, it follows from \(\mathrm{Ric}(\omega_{FS}) = 2\omega_{FS}\) that

\[
(10) \qquad dd^c[\log \xi] = (q - 2) f^* \omega_{FS} - \sum_{j=1}^q (f - a_j)_0 + D_{f, \mathrm{ram}}
+ \mathcal{R} - 2 \sum_{j=1}^q dd^c \log \log \|f, a_j\|^{-2}
\]

in the sense of currents, where \((f - a_j)_0\) is the zero divisor of \(f - a_j\), \(D_{f, \mathrm{ram}}\) is the ramification divisor of \(f\), and \(\mathcal{R} = -dd^c\log h\) is the Ricci form of \(S\). Applying the integral operator

\[
\int_{r_0}^r \frac{dt}{t} \int_{D^{\mathcal{S}}(t)}
\]

to the above identity and using Dynkin formula, we get

\[
\begin{aligned}
\frac{1}{2} \int_{C^{\mathcal{S}}(r)} \log \xi d\pi_o^r &= (q-2) T_f^{\mathcal{S}}(r) - \sum_{j=1}^q N_f^{\mathcal{S}}(r, a_j) + N^{\mathcal{S}}(r, D_{f, \mathrm{ram}}) \\
& \qquad + T^{\mathcal{S}}(r, \mathcal{R}) - \sum_{j=1}^q \int_{C^{\mathcal{S}}(r)} \log \log \|f, a_j\|^{-2} d\pi_o^r + O(1) \\
& \geq (q-2) T_f^{\mathcal{S}}(r) - \sum_{j=1}^q \overline{N}_f^{\mathcal{S}}(r, a_j) + T^{\mathcal{S}}(r, \mathcal{R}) \\
& \qquad - \sum_{j=1}^q \int_{C^{\mathcal{S}}(r)} \log \log \|f, a_j\|^{-2} d\pi_o^r + O(1).
\end{aligned}
\]

The concavity of \(\log\) implies that

\[
(11) \qquad \int_{C^{\mathcal{S}}(r)} \log \log \|f, a_j\|^{-2} d\pi_o^r \leq \log \int_{C^{\mathcal{S}}(r)} \log \|f, a_j\|^{-2} d\pi_o^r
= \log m_f^{\mathcal{S}}(r, a_j) + O(1) \leq \log T_f^{\mathcal{S}}(r) + O(1).
\]

By this, we obtain

\[
(12) \qquad \frac{1}{2} \int_{C^{\mathcal{S}}(r)} \log \xi d\pi_o^r \geq (q-2) T_f^{\mathcal{S}}(r) - \sum_{j=1}^q \overline{N}_f^{\mathcal{S}}(r, a_j) + T^{\mathcal{S}}(r, \mathcal{R}) + O(\log T_f^{\mathcal{S}}(r)) + O(1).
\]

It remains to estimate the upper bound of the term on the left-hand side of \((12)\). By Lemma 3.6, for any \(\delta > 0\)

By uniformization theorem, for each nonnegative constant \(C\), there exists a metric \(h\) such that the Gauss curvature \(K\) satisfies \(-C \leq K \leq 0\). Now, we estimate the lower bound of \(T^{\mathscr{S}}(r, \mathscr{R})\). By Lemmas 2.11 and 2.12, it follows from (8) that

\[
T^{\mathscr{S}}(r, \mathscr{R}) = \frac{1}{2} \int_{D^{\mathscr{S}}(r)} g_r(o, x)K(x) dV(x) - \frac{1}{2} \int_{D^{\mathscr{S}}(r_0)} g_{r_0}(o,x)K(x) dV(x)
\]
\[
\geq -\frac{C}{2} \int_0^r dt \int_{C^{\mathscr{S}}(t)} g_r(o,x) d\sigma_t(x)
\]
\[
= -C \int_0^r dt \int_{C^{\mathscr{S}}(t)} \frac{t}{\| \mathscr{S} \|_h} \log \frac{r}{t} d\pi_o^t
\]
\[
\geq -\frac{C}{\| \mathscr{S} \|_{r,\inf}} \int_0^r t \log \frac{r}{t} dt = -\frac{C r^2}{4 \| \mathscr{S} \|_{r,\inf}},
\]

where

\[
\| \mathscr{S} \|_{r,\inf} = \inf \{ \| \mathscr{S}_x \|_h : |\hat{x}| < r \}.
\]

Combining this with Theorem 3.4, we obtain:

\[
\int_{C^{\mathscr{S}}(r)} \log \xi d\pi_o^r \leq \log \int_{C^{\mathscr{S}}(r)} \xi d\pi_o^r
\]
\[
\leq (1+\delta)^2 \log \int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} \xi dV + \log \| \mathscr{S} \|_{r,\sup} + (2+\delta) \log \gamma(r) + \delta \log r + O(1)
\]
\[
= (1+\delta)^2 \log \int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} f^*\Phi + \log \| \mathscr{S} \|_{r,\sup} + (2+\delta) \log \gamma(r) + \delta \log r + O(1)
\]

holds for all \( r \in (0, R^{\mathscr{S}}) \) outside a set \( E_\delta \) with \( \int_{E_\delta} \gamma(r) dr < \infty \), here \( \gamma \) is an integrable function on \( (0, R^{\mathscr{S}}) \) such that \( \int_0^{R^{\mathscr{S}}} \gamma(r) dr = \infty \). Notice that

\[
\int_{D^{\mathscr{S}}(r)} f^*\Phi = \int_{\mathbb{C}} n_f^{\mathscr{S}}(r, \zeta)\Phi(\zeta)
\]

due to the change of variable formula. Using Fubini theorem, then

\[
\int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} f^*\Phi = \int_{\mathbb{C}} N_f^{\mathscr{S}}(r, \zeta)\Phi(\zeta) \leq c_0 T_f^{\mathscr{S}}(r)
\]

for some positive constant \(c_0\). Combining the above, we conclude that

\[
(13) \quad \int_{C^{\mathscr{S}}(r)} \log \xi d\pi_o^r \leq (1+\delta)^2 \log T_f^{\mathscr{S}}(r) + \log \| \mathscr{S} \|_{r,\sup} + (2+\delta) \log \gamma(r) + \delta \log r + O(1)
\]

holds for all \( r \in (0, R^{\mathscr{S}}) \) outside a set \( E_\delta \) with \( \int_{E_\delta} \gamma(r) dr < \infty \). Put together (12) and (13), then we prove the theorem. □

1. Corollary 3.8. Assume the same conditions as in Theorem 3.4. Suppose, in addition, that \(-C \leq K \leq 0\) for a nonnegative constant \(C\). If \(f\) satisfies

2. then we have the defect relation

3. Theorem 3.7. Assume the same conditions as in Theorem 3.4. Suppose, in addition, that \(-C \leq K \leq 0\) for a nonnegative constant \(C\). Then for any \(\delta > 0\)

4. holds for all \(r \in (0, R^{\mathscr{S}})\) outside a set \(E_\delta\) with \(\int_{E_\delta} \gamma(r) dr < \infty\), where

5. \(\|\mathscr{S}\|_{r,\inf} = \inf \{\|\mathscr{S}_x\|_h : |\hat{x}| < r\}, \quad \|\mathscr{S}\|_{r,\sup} = \sup\{\|\mathscr{S}_x\|_h : |\hat{x}| < r\}\).

6. We recall the definition of the simple defect \(\bar{\delta}_f(a)\) in (3).

7. Let \(\mathscr{R}\) be a compact Riemann surface of genus \(g\). Fix a positive \((1,1)\)-form \(\omega\) on \(\mathscr{R}\). In a local holomorphic coordinate \(\zeta\), we may write \(\omega\) as

8. \[
    \omega = \frac{\sqrt{-1}}{2\pi} w d\zeta \wedge d\bar{\zeta}.
    \]

9. According to Chern's theorem [3], we have that for every \(a \in \mathscr{R}\), there exists a positive function \(u_a\) on \(\mathscr{R}\) such that

10. (14) \quad \(2dd^c[\log u_a] = \omega - \delta_a\),

11. where \(\delta_a\) should be understood as a current. Let \(f : S \to \mathscr{R}\) be a holomorphic mapping. For \(a \in \mathscr{R}\) with \(f \not\equiv a\), by replacing \(\|f, a\|^{-1}\) by \(u_a \circ f\), we define similarly the Nevanlinna's functions of \(f\) as

12. \[
    T_{f,\omega}^{\mathscr{S}}(r) = \int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} f^*\omega,
    \]
    \[
    m_{f,\omega}^{\mathscr{S}}(r, a) = \int_{C^{\mathscr{S}}(r)} \log(u_a \circ f) d\pi_o^r,
    \]
    \[
    N_f^{\mathscr{S}}(r, a) = \int_{r_0}^r \frac{n_f^{\mathscr{S}}(t, a)}{t} dt.
    \]

13. \[
    (q-2) T_f^{\mathscr{S}}(r)
    \leq \sum_{i=1}^q N_f^{\mathscr{S}}(r, a_j) + O\left( \log T_f^{\mathscr{S}}(r) + C r^2 \|\mathscr{S}\|_{r,\inf}^{-1} + \log \|\mathscr{S}\|_{r,\sup} + \log \gamma(r) + \delta \log r \right)
    \]

14. \[
    \limsup_{r \to R^{\mathscr{S}}} \frac{C r^2 \|\mathscr{S}\|_{r,\inf}^{-1} + \log(\gamma(r)\|\mathscr{S}\|_{r,\sup})}{T_f^{\mathscr{S}}(r)} = 0,
    \]

15. \[
    \sum_{j=1}^q \bar{\delta}_f(a_j) \leq 2
    \]

where $K'$ is the Gauss curvature of $\omega$. If $g = 0$, then $\mathscr{R}$ can be regarded as $\mathbb{P}^1(\mathbb{C})$. Since $C_1\omega_{FS} \leq \omega \leq C_2\omega_{FS}$ for two suitable positive constants, then we can confirm the theorem by using the conclusion proved in Theorem 3.4. In the following, we assume that $g \geq 1$. We need to modify the form (9) as

\[
\Phi = \frac{C \omega}{\prod_{j=1}^q u_{a_j}^{-2} \log^2 u_{a_j}^2},
\]

where $C$ is chosen so that $\Phi$ is normalized. Set

\[
f^*\Phi = \xi \frac{\sqrt{-1}}{\pi} h \, dz \wedge d\bar{z}.
\]

Since (14) and (15), then in the sense of currents, (10) becomes

\[
(16) \quad dd^c [\log \xi] = q f^*\omega - f^*(K'\omega) - \sum_{j=1}^q (f-a_j)_0 + D_{f,\text{ram}} + \mathscr{R} - 2 \sum_{j=1}^q dd^c \log \log (u_{a_j}^2 \circ f),
\]

It is similar to (11), we have

\[
\int_{C^{\mathscr{S}}(r)} \log\log u_{a_j}^2 \, d\pi_o^r \leq \log T_{f,\omega}^{\mathscr{S}}(r) + O(1).
\]

Integrating both sides of (16) and using Dynkin formula, we get

\[
\frac{1}{2} \int_{C^{\mathscr{S}}(r)} \log \xi \, d\pi_o^r
\geq q T_{f,\omega}^{\mathscr{S}}(r)
- \int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(t)} f^*(K'\omega)
- \sum_{j=1}^q N_f^{\mathscr{S}}(r, a_j)
+ T^{\mathscr{S}}(r, \mathscr{R})
+ O(\log T_{f,\omega}^{\mathscr{S}}(r)) + O(1).
\]

For the second term on the right-hand side of the above identity, we use the change of variable formula and Gauss–Bonnet formula to yield that

\[
\int_{r_0}^r \frac{dt}{t} \int_{D^{\mathscr{S}}(r)} f^*(K'\omega) = \int_{r_0}^r \frac{dt}{t} \int_{\mathscr{R}} n_f^{\mathscr{S}}(t, \zeta) K'(\zeta) \omega(\zeta)
\]
\[
= \int_{\mathscr{R}} \left[ \int_{r_0}^r \frac{n_f^{\mathscr{S}}(t, \zeta)}{t} dt \right] K'(\zeta) \omega(\zeta)
\]
\[
= \int_{\mathscr{R}} N_f^{\mathscr{S}}(r, \zeta) K'(\zeta) \omega(\zeta)
\]
\[
\geq \int_{\mathscr{R}} (T_{f,\omega}^{\mathscr{S}}(r) + O(1)) K'(\zeta) \omega(\zeta)
\]
\[
= (2 - 2g) T_{f,\omega}^{\mathscr{S}}(r) + O(1).
\]

Thus

\[
(17) \quad \frac{1}{2} \int_{C^{\mathscr{S}}(r)} \log \xi \, d\pi_o^r \geq (q - 2 + 2g) T_{f,\omega}^{\mathscr{S}}(r) - \sum_{j=1}^q \bar{N}_f^{\mathscr{S}}(r, a_j)
\]
\[
\qquad\qquad\qquad + T^{\mathscr{S}}(r, \mathscr{R}) + O(\log T_{f,\omega}^{\mathscr{S}}(r)) + O(1).
\]

On the other hand, it follows by using similar arguments as in the derivation of (13) that

\[
(18) \quad \int_{C^{\mathscr{S}}(r)} \log \xi \, d\pi_o^r \leq (1+\delta)^2 \log T_{f,\omega}^{\mathscr{S}}(r) + \log \|\mathscr{S}\|_{r,\sup}
\]
\[
\qquad\qquad\qquad + (2+\delta) \log \gamma(r) + \delta \log r + O(1)
\]

holds for all \( r \in (0, R^{\mathscr{S}}) \) outside a set \( E_\delta \) with \( \int_{E_\delta} \gamma(r) dr < \infty \). Put together (17) and (18), then we conclude the following Second Main Theorem.

## Theorem 4.1.
Let \( (S, h; \mathscr{S}) \) be a \( \mathscr{S} \)-exhausted Hermitian Riemann surface of \( \mathscr{S} \)-radius \( R^{\mathscr{S}} \) with respect to \( o \), and \( \mathscr{R} \) be a compact Riemann surface of genus \( g \). Fix a positive (1,1)-form \( \omega \) on \( \mathscr{R} \). Let \( \gamma \) be an integrable function on \( (0, R^{\mathscr{S}}) \) with \( \int_0^{R^{\mathscr{S}}} \gamma(r) dr = \infty \). Let \( f: S \to \mathscr{R} \) be a nonconstant holomorphic mapping and \( a_1, \dots, a_q \) be distinct points in \( \mathscr{R} \). Then for any \( \delta > 0 \)

\[
(q-2+2g) T_{f,\omega}^{\mathscr{S}}(r) + T^{\mathscr{S}}(r, \mathscr{R})
\]
\[
\leq \sum_{j=1}^q \bar{N}_f^{\mathscr{S}}(r, a_j) + O\left(\log T_{f,\omega}^{\mathscr{S}}(r) + \log \|\mathscr{S}\|_{r,\sup} + \log \gamma(r) + \delta \log r\right)
\]

holds for all \( r \in (0, R^{\mathscr{S}}) \) outside a set \( E_\delta \) with \( \int_{E_\delta} \gamma(r) dr < \infty \), where

\[
\|\mathscr{S}\|_{r,\sup} = \sup\{ \|\mathscr{S}_x\|_h : |\hat{x}| < r \}.
\]

Similar to Theorem 3.7, we have:

Theorem 4.2. Assume the same conditions as in Theorem 4.1. Suppose, in addition, that −C ≤ K ≤ 0 for a nonnegative constant C. Then for any δ > 0

holds for all r ∈ (0, R^S) outside a set E_δ with ∫_{E_δ}γ(r) dr < ∞, where

‖S‖_{r,inf} = inf{‖S_x‖_h : |x̂| < r}, ‖S‖_{r,sup} = sup{‖S_x‖_h : |x̂| < r}.

Theorem 4.2 derives a defect relation, i.e., Theorem III in Introduction.

The authors are grateful to the referee for valuable comments on this paper.

74                 XIANJING DONG AND SHUANGSHUANG YANG

[13] M. Păun and N. Sibony, "Value distribution theory for parabolic Riemann surfaces", 2014. arXiv 1403.6596

[14] M. Ru, Nevanlinna theory and its relation to diophantine approximation, World Scientific Publishing Co., River Edge, NJ, 2001. MR Zbl

[15] M. Ru and N. Sibony, "The second main theorem in the hyperbolic case", Math. Ann. 377:1-2 (2020), 759–795. MR Zbl

[16] B. V. Shabat, Distribution of values of holomorphic mappings, Translations of Mathematical Monographs 61, American Mathematical Society, Providence, RI, 1985. MR Zbl

[17] H.-H. Wu, The equidistribution theory of holomorphic curves, Annals of Mathematics Studies 64, Princeton University Press, Princeton, N.J., 1970. MR Zbl

Received January 4, 2022. Revised April 26, 2022.

XIANJING DONG
SCHOOL OF MATHEMATICAL SCIENCES
QUFU NORMAL UNIVERSITY
QUFU
CHINA
xjdong05@126.com

SHUANGSHUANG YANG
DEPARTMENT OF MATHEMATICS
NANJING UNIVERSITY
NANJING
CHINA
ssyang97@126.com

Daryl Cooper
Department of Mathematics
University of California
Santa Barbara, CA 93106-3080
cooper@math.ucsb.edu

Jiang-Hua Lu
Department of Mathematics
The University of Hong Kong
Pokfulam Rd., Hong Kong
jhlu@maths.hku.hk

Wee Teck Gan
Mathematics Department
National University of Singapore
Singapore 119076
matgwt@nus.edu.sg

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July 2022
Volume 319
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