j. differential geometry

58 (2001) 309-329

HÖLDER ESTIMATES AND REGULARITY FOR HOLOMORPHIC AND HARMONIC FUNCTIONS

PETER LI & JIAPING WANG

Abstract

In this paper, we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions. Moreover, if we further assume that the volume decays at most polynomially at a singular point, then we obtain a Hölder estimate of the holomorphic function at that point. In a similar spirit, we also established a continuity estimate for bounded harmonic functions, with a finite dimensional exception, at a singular point of a manifold satisfying a weak mean value property and a polynolmial volume decay condition.

0. Introduction

The theory of DeGiorgi-Nash-Moser asserts that if f ≥ 0 is a nonconstant solution to the partial differential equation

Lf = 0

on a ball B(2R) ⊂ ℝ^{n}, with

L = \frac{∂}{∂x_{i}} (a_{ij} \frac{∂}{∂x_{j}} )

being a uniformly elliptic operator of divergence form, then f must satisfy the Harnack inequality

sup_{B(R)} f ≤ C inf_{B(R)} f

The first author was partially supported by NSF grant #DMS-9971418; the second author was partially supported by NSF grant #DMS-0072181.

Received June 19, 2001.

for some constant C depending only on the ellipticity bounds of (a_{ij}) and the dimension n. An upshot of the Harnack inequality is the oscillation estimate, which states that there exist constants C > 0 and 0 < α ≤ 1, such that if

ω(r) = sup_{B(r)} f − inf_{B(r)} f

denotes the oscillation of f over B(r), then

(0.1)  ω(r) ≤ C (\frac{r}{R})^{α} ω(R).

In particular, by letting r → 0, this implies the C^{α} regularity of f.

On the other hand, if f is defined on the whole ℝ^{n}, by letting R → ∞ while keeping r fixed, this implies f must be constant if f satisfies the growth condition

|f(x)| = o(|x|^{α}).

For simplicity, let us refer this property as the α-Liouville property. One can think of the α-Liouville property as the dual of the C^{α} regularity, or perhaps C^{α} regularity at ∞.

Taking this on a manifold setting, by unraveling Moser’s original argument [16, 4], one notes that the manifold only needs to satisfy a Poincaré inequality, a Sobolev inequality, and volume doubling property in order for Moser’s program to go through. The essence was further illuminated by the works of Saloff-Coste [14, 15] and Grigor’yan [3], that a Poincaré inequality and a volume doubling property suffice to imply the Harnack inequality, hence the α-Liouville property and the C^{α} regularity.

In [5], the first author proved if a manifold satisfies a mean value inequality and a volume comparison condition, then the space of harmonic functions that grow at most polynomially of degree d must be finite dimensional. Later in [7], the authors gave an even weaker hypothesis to ensure the finite dimensionality for harmonic functions. In fact, the arguments in [5] and [7] are also valid for solutions of uniformly elliptic operators of both divergence and non-divergence form. The authors showed that if a manifold M satisfies a weak mean value property (Definition 1) and has polynomial volume growth, then the space of harmonic functions on M satisfying the growth condition

|f(x)| = O(r^{d}(x))

for some d > 0 must be finite dimensional. Consequently, as observed in [7], if the manifold is in addition Kähler, then there are no nonconstant bounded holomorphic functions defined on M. This hinges on the simple fact that powers of a holomorphic function are also holomorphic. Note that since the Sobolev inequality alone implies both the weak mean value property by Moser’s iteration and the polynomial volume growth [7], one concludes that the space of polynomial growth harmonic functions of fixed degree must be finite dimensional on any complete manifold satisfying Sobolev inequality. They also addressed the issues of sharpness of the dimension estimate in their subsequent papers [8] and [9].

On the other hand, if C^{α} regularity is dual to the α-Liouville property, then one may ask if any local regularity property can be asserted from the finite dimensionality of the space of polynomial growth harmonic functions of fixed degree. A simple case to examine is the holomorphic function case where we have the convenience of taking powers. The purpose of this paper is to prove (Theorem 4) that if a complete Kähler manifold satisfies the weak mean value property then an estimate of the type (0.1) holds with the constant C depending on the ratio of the volumes of balls. Moreover, if at a point the manifold has at most polynomial volume decay for small balls, then it will have C^{α} regularity (Corollary 5). On the other hand, if the manifold has at most polynomial volume growth for large balls, then it has the α- Liouville property (Corollary 8) for holomorphic functions. In fact, we will obtain a uniform C^{α} estimate for all holomorphic functions (Corollary 5) if we assume a volume comparison condition.

A typical situation to apply Theorem 4 is when the Kähler manifold is a singular algebraic or minimal variety. In this case, the subvariety inherits a Sobolev inequality from the ambient manifold [11] hence possesses the weak mean value property. However, in order to deal with this situation, the balls in Theorem 4 are usually taken to be extrinsic balls rather than geodesic balls. All the argument in Theorem 4 will carry through with the one exception that extrinsic balls are not necessarily connected. In order to overcome this, the assumption of local irreducibility (Definition 6) is imposed at the point in question. In fact, the local irreducibility assumption is necessary as can be seen by taking M as the union of the z and w planes in ℂ^{2}. The singular point p = (0, 0) disconnects the two copies of ℂ hence violates the local irreducibility assumption. Moreover, the function given by 1 on the z-plane and 0 on the w-plane will be bounded holomorphic function on M that is discontinuous.

Also note that since any complete manifold with Ricci curvature bounded from below possesses the weak mean value property locally [6], one can also interpret the local C^{α} estimate as a weaker form of Schwarz lemma (see [17], [2], and [13]) on these singular spaces.

In the case of harmonic functions, since it is possible to have a finite dimensional space of nonconstant bounded harmonic functions on a complete manifold with the weak mean value property and polynomial volume growth, we do not expect to have a C^{α} regularity theory which holds for all harmonic functions. However, we do expect that for each singular point p of M, other than a finite dimensional exception, all harmonic functions are C^{α}. While this is the ideal local analog for the global theory, we only managed to prove this theorem for C^{0} regularity in Theorem 9. We would like to point out that the local irreducibility assumption is not necessary in Theorem 9. The finite dimension exception allows some disconnect components for the complement of the singular set of M. Moreover, the mean value constant λ also puts a restriction on the number of disconnected components.

In the last section, we proved a finite dimensionality theorem on the space

H_{d}(L, M) = {f | Lf = 0 and |f|(x) = O(r^{d}(x))}

of L-harmonic functions with at most polynomial growth of degree d defined on a manifold with nonnegative Ricci curvature. The operator L is an elliptic operator of divergence form with

Lf = div(A(x)∇f)

where A : T M → T M is an endomorphism of the tangent bundle. The assumption on A(x) is given by measurable coefficients in local coordinates and also

0 < ϕ(r) I ≤ A(x) ≤ Φ(r) I < ∞

for all x ∈ B_{p}(\frac{3r}{2} ) \ B_{p}(\frac{r}{2}) with

\frac{Φ(r)}{ϕ(r)} ≤ Ω

for some fixed constant Ω > 0. Note that this assumption on A allows the operator to grow or decay polynomially as long as the lower bound and the upper bound are of the same order. Examples of these is when

A(x) = r^{t}(x) A_{0}

for some −∞ < t < ∞.

We would like to point out that when A is uniformly bounded on the entire space and further assuming that the operator has an asymptotic limit, the issue of finding the dimension of H_{d}(L, ℝ^{n}) was studied by Avellaneda-Lin [1], Moser-Struwe [12], and Lin [10]. For general uniformly elliptic operators of divergence and non-divergence form, the issue of finite dimensionality of H_{d}(L, M) and to estimate its dimension was studied by the first author in [5] and then by the authors in [7] and [9]. Theorem 10 is the first situation which the coefficients are not required to be uniformly bounded over the whole manifold.

1. Preliminaries

Let us first recall the weak mean value property.

Definition 1. A Riemannian manifold M is said to satisfy the weak mean value property WM(R) if there exist constants b ≥ 1 and λ ≥ 1 such that

u^{2}(q) ≤ \frac{λ}{V_{q}(r)} ∫_{B_{q}(br)} u^{2}

for all r ≤ R and subharmonic function u defined on the geodesic ball B_{q}(bR), where V_{q}(r) denotes the volume of the ball B_{q}(r).

As pointed out before, a complete manifold with Ricci curvature bounded from below possesses the weak mean value property WM(R) for any fixed R [6]. More generally, Moser’s iteration implies that a complete manifold M possesses the weak mean value property WM(R) if the following type of Sobolev inequality S(R, ν) is valid on M:

(∫_{B_{p}(r)} |f|^{\frac{2ν}{ν−2}} )^{\frac{ν−2}{ν}} ≤ B V_{p}^{− \frac{2}{ν}} (r) ∫_{B_{p}(r)}(r^{2} |∇f|^{2} + f^{2})

for all p ∈ M, 0 < r ≤ R and for all f ∈ H_{0}^{1}(B_{p}(r)), where B > 0 and ν > 2 are constants.

The following lemma is a simplified version of a lemma used in [5]. It is a key ingredient in most of the finite dimensionality proofs.

Lemma 2. Let M be a Riemannian (not necessarily complete) manifold satisfying the weak mean value property WM(R). Let K be a k-dimensional linear space of complex functions on M with the property that

Δ|u|^{2} ≥ 0

for all u ∈ K. If we denote by

A_{t}(u, v) = ∫_{B_{p}(t)} uv

the Hermitian inner product on K, then the trace of A_{r} with respect to the inner product A_{(2b+1)r} must satisfy

tr_{A_{(2b+1)r}}A_{r} ≤ λ

for all r ≤ \frac{R}{2}.

Proof. Let {f_{1}, …, f_{k}} be a unitary basis with respect to the Hermitian inner product A_{(2b+1)r}. For any ϵ > 0 sufficiently small, let q ∈ B_{p}(r) be a point such that

(1.1)  (1 + ϵ) ∑_{i=1}^{k} |f_{i}|^{2}(q) ≥ sup_{B_{p}(r)} ∑_{i=1}^{k} |f_{i}|^{2}.

By a unitary transformation if necessary, we may assume that f_{i}(q) = 0 for all i ≥ 2 and

∑_{i=1}^{k} |f_{i}|^{2}(q) = |f_{1}|^{2}(q).

Applying the weak mean value property, we have

(1.2)  tr_{A_{(2b+1)r}}A_{r} = ∫_{B_{p}(r)} ∑_{i=1}^{k} |f_{i}|^{2}

≤ V_{p}(r) sup_{B_{p}(r)} ∑_{i=1}^{k} |f_{i}|^{2}

≤ V_{p}(r) (1 + ϵ) |f_{1}|^{2}(q)

≤ λ (1 + ϵ) V_{p}(r) V_{q}(2r)^{−1} ∫_{B_{q}(2br)} |f_{1}|^{2}

≤ λ (1 + ϵ) ∫_{B_{p}((2b+1)r)} |f_{1}|^{2}

= λ (1 + ϵ).

The lemma follows by letting ϵ → 0. q.e.d.

Let us remark that when M is a complete, smooth Riemannian manifold, then the maximum principle asserts that there exists q ∈ ∂B_{p}(r) such that

(1.3)  ∑_{i=1}^{k} |f_{i}|^{2}(q) = sup_{B_{p}(r)} ∑_{i=1}^{k} |f_{i}|^{2}.

In this case, we can replace (1.1) by (1.3). Moreover, if we replace (1.2) by

tr_{A_{\frac{3r}{2}}} A_{r} = ∫_{B_{p}(r)} ∑_{i=1}^{k} |f_{i}|^{2}

≤ V_{p}(r) sup_{B_{p}(r)} ∑_{i=1}^{k} |f_{i}|^{2}

= V_{p}(r) |f_{1}|^{2}(q)

≤ λ V_{p}(r) V_{q}(\frac{r}{2b})^{−1} ∫_{B_{q}( \frac{r}{2} )} |f_{1}|^{2}

≤ λ V_{p}(r) V_{q}(\frac{r}{2b})^{−1} ∫_{B_{p}( \frac{3r}{2} )} |f_{1}|^{2},

we conclude that

(1.4)  tr_{A_{\frac{3r}{2}}} A_{r} ≤ \frac{λ V_{p}(r)}{V_{q}(\frac{r}{2b})}

where we only need to assume that the weak mean value inequality

(1.5)  u^{2}(q) ≤ \frac{λ}{V_{q}(\frac{r}{2b})} ∫_{B_{q}( \frac{r}{2} )} u^{2}

is valid for points q which is distance r from p. This observation allows us to consider elliptic operators of divergence form that may not have uniformly bounded coefficients. We will address this issue in §5.

Lemma 3. Let M be a Riemannian (not necessarily complete) manifold satisfying the weak mean value property WM(R). Let K be a k-dimensional linear space of complex functions on M with the property that

Δ|u|^{2} ≥ 0

for all u ∈ K. If k > λ and if

A_{t}(u, v) = ∫_{B_{p}(t)} uv

denotes the Hermitian inner product on K, then the determinant of A_{r} with respect to the inner product A_{R} must satisfy

det_{A_{R}}A_{r} ≤ (\frac{k r^{β}}{λ R^{β}} )^{k}

for all r ≤ R, where

β = \frac{ln k − ln λ}{ln(2b + 1)}.

Proof. Lemma 2 and the arithmetic-geometric mean inequality assert that

det_{A_{(2b+1)r}}A_{r} ≤ (\frac{tr_{A_{(2b+1)r}}A_{r}}{k} )^{k}

≤ (\frac{λ}{k})^{k} .

Setting γ = (2b + 1)^{−1} and r = γ^{i}R for i = 0, 1, …, we can rewrite this as

det_{A_{γ^{i}R}}A_{γ^{i+1}R} ≤ (\frac{λ}{k})^{k} .

Iterating this inequality j times by letting i = 0, …, j − 1, we have

det_{A_{R}}A_{γ^{j}R} ≤ (\frac{λ}{k})^{jk} .

In particular, we have

det_{A_{R}}A_{r} ≤ (\frac{r}{R})^{\frac{k(ln λ−ln k)}{ln γ}}

= (\frac{r}{R})^{\frac{k(ln k−ln λ)}{ln(2b+1)}}

if r = γ^{j}R for some j = 1, 2, …. When γ^{j+1}R < r < γ^{j}R, we observe that

det_{A_{R}}A_{r} = det_{A_{R}}A_{γ^{j}R} det_{A_{γ^{j} R}}A_{r}.

By choosing an unitary basis {f_{1}, …, f_{k}} with respect to A_{γ^{j}R} that diagonalizes A_{r}, we note that

det_{A_{γ^{j} R}}A_{r} = ∏_{i=1}^{k} ∫_{B_{p}(r)} |f_{i}|^{2}

≤ ∏_{i=1}^{k} ∫_{B_{p}(γ^{j}R)} |f_{i}|^{2}

= 1.

Hence

det_{A_{R}}A_{r} ≤ det_{A_{R}}A_{γ^{j}R}

≤ (\frac{λ}{k})^{jk} .

Using the fact that γ^{j+1}R < r and assuming that k > λ, we get

(\frac{λ}{k})^{jk} ≤ (\frac{k}{λ})^{k} (\frac{r}{R})^{\frac{k(ln k−ln λ)}{ln(2b+1)}} ,

which implies that

det_{A_{R}}A_{r} ≤ (\frac{k r^{\frac{(ln k−ln λ)}{ln(2b+1)}}}{λ R^{\frac{(ln k−ln λ)}{ln(2b+1)}}} )^{k}

for all r ≤ R. q.e.d.

2. Holomorphic Functions

We are now ready to prove the Hölder estimate for holomorphic functions.

Theorem 4. Let M be a Kähler (not necessarily complete) manifold satisfying the weak mean value property WM(R). Suppose f is a holomorphic function defined on the geodesic ball B_{p}(R). For all r ≤ R and for all k + 1 > λ, if we denote ω(r) to be the oscillation of f on B_{p}(r), then

ω(r) ≤ Ck (\frac{V_{p}(R)}{V_{p}(r)} (\frac{r}{R})^{β})^{\frac{1}{k(k+1)}} ω(R),

where C > 0 is a constant depending on λ and b alone, and

β = \frac{ln(k + 1) − ln λ}{ln(2b + 1)}.

Proof. We may assume that the function f is nonconstant and normalized to have ω(R) = 1, inf_{B_{p}(R)} |f| = 0, and sup_{B_{p}(R)} |f| ≤ 1. Let us consider the linear space K spanned by the functions {1, f, f^{2}, …, f^{k}}. Then K is of complex dimension k + 1. For h ∈ K with A_{R}(h, h) = 1, there exist complex numbers c_{0}, c_{1}, …, c_{k} such that h = ∑_{i=0}^{k} c_{i}f^{i}. Hence

1 = A_{R}(h, h)

= ∫_{B_{p}(R)} (∑_{i=0}^{k} c_{i}f^{i}) (\overline{∑_{i=0}^{k} c_{i}f^{i}})

≤ ∫_{B_{p}(R)} (∑_{i=0}^{k} |c_{i}|)^{2}

≤ V_{p}(R) (∑_{i=0}^{k} |c_{i}|)^{2} ,

and

(2.1)  (∑_{i=0}^{k} |c_{i}|)^{2} ≥ \frac{1}{V_{p}(R)}.

On the other hand, we can choose points z_{0}, z_{k} ∈ B_{p}(r) such that

|f(z_{0}) − f(z_{k})| ≥ \frac{ω(r)}{2}.

Since f(B_{p}(r)) is connected in the complex plane ℂ, for any s ≤ |f(z_{0})− f(z_{k})|, we have

∂B_{f(z_{0})}(s) ∩ f(B_{p}(r)) ≠ ∅.

Therefore, for i = 1, …, k − 1, we can choose point z_{i} ∈ B_{p}(r) such that

f(z_{i}) ∈ ∂B_{f(z_{0})} (\frac{i}{k} |f(z_{0}) − f(z_{k})|) .

hölder estimates and regularity 319

In particular, we have

(2.2)  |f(z_{i}) − f(z_{j})| ≥ \frac{ω(r)}{2k}

for any i ≠ j with i, j = 0, 1, …, k.

Now consider the linear system

∑_{i=0}^{k} c_{i}f^{i}(z_{j}) = h(z_{j})

for j = 0, …, k. Let A denote the matrix

A = (\begin{matrix}1 & f(z_{0}) & … & f^{k}(z_{0}) \\ 1 & f(z_{1}) & … & f^{k}(z_{1}) \\ . & . & . \\ 1 & f(z_{k}) & … & f^{k}(z_{k}) \end{matrix}) .

Then the linear system can be written as

A (c_{0} ; c_{1} ; ⋮ ; c_{k}) = (h(z_{0}) ; h(z_{1}) ; ⋮ ; h(z_{k})) ,

and

(c_{0} ; c_{1} ; ⋮ ; c_{k}) = A^{−1} (h(z_{0}) ; h(z_{1}) ; ⋮ ; h(z_{k})) .

Cramer’s rule asserts that the (ij)^{th} entry of A^{−1} is given by

|(A^{−1})_{ij}| = |\frac{detM_{ij}}{detA} | ,

where M_{ij} is the (ij)^{th} minor of A. Since |f| ≤ 1, we have

|detM_{ij}| ≤ k!.

Also, by (2.2),

|detA| = ∏_{0≤i<j≤k} |f(z_{i}) − f(z_{j})|

≥ (\frac{ω(r)}{2k})^{\frac{k(k+1)}{2}} .

320 peter li & jiaping wang

If we let g(r) = sup_{B_{p}(r)} |h|, then

|c_{i}| ≤ \frac{(k + 1) k! (2k)^{\frac{k(k+1)}{2}}}{ω(r)^{\frac{k(k+1)}{2}}} g(r).

Hence we conclude that

∑_{i=0}^{k} |c_{i}| ≤ \frac{(k + 1)^{2} k! (2k)^{\frac{k(k+1)}{2}}}{ω(r)^{\frac{k(k+1)}{2}}} g(r).

Combining with (2.1), we obtain the estimate

\frac{1}{V_{p}(R)} ≤ \frac{(k + 1)^{4} (k!)^{2} (2k)^{k(k+1)}}{ω(r)^{k(k+1)}} g^{2}(r),

hence

(2.3)  g^{2}(r) ≥ V_{p}(R)^{−1} (k + 1)^{−4} (k!)^{−2} (2k)^{−k(k+1)} ω(r)^{k(k+1)}.

On the other hand,

g^{2}(r) ≤ 2|h(q)|^{2}

for some q ∈ B_{p}(r) and by the weak mean value property, we have

g^{2}(r) ≤ 2|h(q)|^{2}

≤ \frac{2λ}{V_{q}(2r)} ∫_{B_{q}(2br)} |h|^{2}

≤ \frac{2λ}{V_{q}(2r)} ∫_{B_{p}((2b+1)r)} |h|^{2}.

Thus, applying to (2.3), we conclude that

∫_{B_{p}((2b+1)r)} |h|^{2} ≥ (2λ)^{−1} V_{q}(2r) g^{2}(r) (2.4)

≥ (2λ)^{−1} (k + 1)^{−4} (k!)^{−2} (2k)^{−k(k+1)} ω(r)^{k(k+1)} \frac{V_{p}(r)}{V_{p}(R)}.

Since h ∈ K with A_{R}(h, h) = 1 is arbitrary, (2.4) implies that

det_{A_{R}}A_{(2b+1)r} ≥ (2λ)^{−(k+1)} (k + 1)^{−4(k+1)} (k!)^{−2(k+1)} (2k)^{−k(k+1)^{2}} (2.5)

· ω(r)^{k(k+1)^{2}} (\frac{V_{p}(r)}{V_{p}(R)})^{k+1} .

On the other hand, Lemma 3 asserts that

det_{A_{R}}A_{(2b+1)r} ≤ (\frac{(k + 1) ((2b + 1)r)^{β}}{λ R^{β}} )^{k+1} .

Hence combining with (2.5), we have

ω(r) ≤ (2^{k^{2}+k+1} (k + 1)^{5} (k!)^{2} k^{k(k+1)} (2b + 1)^{β} (\frac{r}{R})^{β} \frac{V_{p}(R)}{V_{p}(r)})^{\frac{1}{k(k+1)}}

≤ C k (\frac{V_{p}(R)}{V_{p}(r)} (\frac{r}{R})^{β} )^{\frac{1}{k(k+1)}} ,

where

β = \frac{ln(k + 1) − ln λ}{ln(1 + 2b)}.

This proves the theorem. q.e.d.

3. Applications

We will now apply the estimate of Theorem 4 to obtain Hölder estimate and α-Liouville property for holomorphic functions.

Corollary 5. Let M be a Kähler (not necessarily complete) manifold satisfying the weak mean value property WM(R). Suppose there exist constants C_{0}(p, R) > 0 and ν > 0 such that

C_{0}(p, R) V_{p}(R) ≤ (\frac{R}{r})^{ν} V_{p}(r)

for all r < R. Then there exists α = α(ν, λ, b) depending only on the quantities ν, λ, and b such that if f is a holomorphic function defined on B_{p}(R) then its oscillation ω(r) must satisfy

ω(r) ≤ c(λ, b, C_{0}(p, R), ν) ω(R) (\frac{r}{R})^{α} .

In particular, if the constant C_{0} is independent of p and R, that is, there exist constants C_{1} > 0 and ν > 0 such that

C_{1} V_{p}(R) ≤ (\frac{R}{r})^{ν} V_{p}(r)

for all p ∈ M and r < R, then we obtain the uniform estimate

ω(r) ≤ c(λ, b, C_{1}, ν) (\frac{r}{R})^{α} ω(R).

Proof. Applying the volume lower bound to Theorem 4, we have

(3.1)  ω(r) ≤ Ck (\frac{r^{β−ν}}{C_{0} R^{β−ν}} )^{\frac{1}{k(k+1)}} ω(R).

By choosing k sufficiently large such that β ≥ 2ν this implies that

ω(r) ≤ c(λ, b, C_{0}, ν) (\frac{r}{R})^{α} ω(R).

This proves the first part of the corollary. The second part of the corollary follows immediately. q.e.d.

Observe that the above argument can be applied to the situation when M is a submanifold of another Riemannian manifold N. In this case, one will replace the intrinsic distance function by the extrinsic distance function. Moreover, the geodesic balls should be replaced by extrinsic balls obtained by taking geodesic balls of N intersecting M. Particular cases which the theorem may apply are complex subvarieties of a Kähler manifold or minimal submanifolds which are Kähler. In these cases, the Kähler variety M may be singular with the singular set given by Σ. In this situation, the theorem will apply to the incomplete manifold M \ Σ. Due to the fact that we are considering the extrinsic ball, the set B_{p}(R) \ Σ may not be connected, hence a connectedness assumption is required.

Definition 6. A singular submanifold M with singular set Σ is said to be locally irreducible if for any extrinsic ball B_{p}(r) with sufficiently small radius r, the set B_{p}(r) \ Σ is connected.

Corollary 7. Let M be a locally irreducible Kähler subvariety of a complete manifold N. Suppose M satisfies the weak mean value property WM(R) and the volume of extrinsic balls satisfy

C_{0} r^{α} ≤ V_{p}(r)

for some constants C_{0} > 0 and α > 0, and for all r ≤ R. Suppose f is a bounded holomorphic function defined on the extrinsic ball B_{p}(R) with p ∈ Σ, then f must be Hölder continuous at the point p.

Note that when M is an algebraic variety, it is known that bounded holomorphic functions can be extended across the singular set if M is irreducible and locally irreducible. In this case, the variety inherits a Sobolev inequality from ℂℙ^{N}, hence satisfies the weak mean value inequality. Moreover, the volume comparison condition is also satisfied with the constant involving the multiplicity of the variety. Hence Corollary 7 can be viewed as a generalization of this theorem.

Corollary 8. Let M be a Kähler manifold satisfying weak mean value property WM(R) for all R > 0. Suppose that M has polynomial volume growth of order ν. Then there exists a positive number α such that every holomorphic function on M of growth order at most α must be constant.

Proof. Since M has polynomial volume growth of order ν, we have

V_{p}(R) ≤ C_{2} V_{p}(1) R^{ν}

for all R ≥ 1. Let f be a holomorphic function on M and ω(r) its oscillation on the ball B_{p}(r). Then by Theorem 4, we conclude that there exists α > 0 such that

ω(1) ≤ c R^{−α} ω(R)

for all R ≥ 1. In particular, if f is of growth order strictly less than α, then by letting R go to infinity we get ω(1) = 0. Thus, f must be constant by the unique continuation property of the holomorphic functions. q.e.d.

4. Continuity of Harmonic Functions

Although we are still unsuccessful in proving Hölder continuity for the real case — harmonic functions — we manage to prove that for any given point, other than a finite dimensional exception, all harmonic functions are continuous.

Theorem 9. Let M be a singular Riemannian manifold satisfying the weak mean value property WM(R). Let us denote the singular set of M by Σ. Let p ∈ Σ be a fixed point with the property that the volume of the ball of radius r centered at p satisfies

V_{p}(r) ≥ C r^{ν}

for some constants C, ν > 0. Then there exists a k-dimensional subspace H of bounded harmonic functions defined on B_{p}(1) with

k ≤ λ (2b + 1)^{ν},

such that if f is a bounded harmonic function defined on the geodesic ball B_{p}(1), then f − ϕ is continuous at p for some ϕ ∈ H.

Proof. To prove the theorem, it suffices to show that for any subspace K of bounded harmonic functions defined on B_{p}(1), with dimension k > λ (2b + 1)^{ν}, there must have at least one nonzero function f ∈ K such that f is continuous at p. Arguing by contradiction, if this is not the case, then for each f ∈ K, there exists ϵ_{f} > 0, such that the oscillation ω_{f}(r) of f on the ball B_{p}(r) must satisfy

ω_{f}(r) ≥ ϵ_{f}

for all r. Observe that since ω_{f}(r) is a monotonic non-increasing function of r, ϵ_{f} can be taken to be

ϵ_{f} = inf_{r→0} ω_{f}(r).

We now claim that ϵ_{f} > 0 is a continuous function defined on the L^{2}(B_{p}(1))-unit sphere 𝕊(K) of K. Then the fact that K is finite dimensional implies the set 𝕊(K) is compact and we can choose

ϵ = inf_{f∈𝕊(K)}ϵ_{f} > 0

such that

ω_{f}(r) ≥ ϵ

for all r ≤ 1 and for all f ∈ 𝕊(K).

To prove the continuity of ϵ_{f}, we consider the function

f_{t} = f cos t + h sin t

where f and h are orthogonal in 𝕊(K). Obviously f_{t} ∈ 𝕊(K) and f_{t} → f as t → 0. Then for x, y ∈ B_{p}(r) and 0 ≤ t ≤ π/2, we have

f_{t}(x) − f_{t}(y) = (f(x) − f(y)) cos t + (h(x) − h(y)) sin t
≤ ω_{f}(r) cos t + (h(x) − h(y)) sin t.

However, the mean value inequality implies that

h^{2}(x) ≤ \frac{λ}{V_{x}(b^{−1}(1 − r))} ∫_{B_{x}((1−r))} h^{2}

≤ \frac{λ}{V_{x}(b^{−1}(1 − r))} ∫_{B_{p}(1)} h^{2}

= \frac{λ}{V_{x}(b^{−1}(1 − r))}.

Since the same estimate applies to the point y, we conclude that

ω_{f_{t}}(r) ≤ ω_{f}(r) cos t + C ( inf_{x∈B_{p}(r)} V_{x}(b^{−1}(1 − r)))^{− \frac{1}{2}} sin t.

Letting r → 0, we have

ϵ_{f_{t}} ≤ ϵ_{f} cos t + C (V_{p}(b^{−1}))^{− \frac{1}{2}} sin t.

Obviously, by reversing the role of f and f_{t}, we see that ϵ_{f_{t}} → ϵ_{f} as t → 0, which proves the continuity of ϵ_{f} and the existence of ϵ.

The above argument also proved that

ω_{f} (\frac{r}{4b}) ≤ C ( inf_{x∈B_{p}(\frac{r}{4b})} V_{x} (\frac{3r}{4b} ))^{− \frac{1}{2}} (∫_{B_{p}(r)} f^{2})^{\frac{1}{2}}

≤ C V_{p}^{− \frac{1}{2}} (\frac{r}{2b}) (∫_{B_{p}(r)} f^{2})^{\frac{1}{2}} .

In particular,

(4.1)  ϵ^{2} ≤ C V_{p}^{−1} (\frac{r}{2b}) ∫_{B_{p}(r)} f^{2}

for all f ∈ 𝕊(K). However, according to Lemma 3, we have

det_{A_{1}}A_{r} ≤ (\frac{k r^{β}}{λ} )^{k}

for β = \frac{ln k−ln λ}{ln(2b+1)}. By choosing an A_{1} orthonormal basis \{f_{1}, …, f_{k}\}, which diagonalizes A_{r}, we have

∏_{i=1}^{k} ∫_{B_{p}(r)} f_{i}^{2} ≤ (\frac{k r^{β}}{λ} )^{k} .

Combining with (4.1), we conclude that

ϵ^{2k} ≤ C^{k} V_{p}^{−k} (\frac{r}{2b}) (\frac{k r^{β}}{λ} )^{k} .

Using the assumption on the lower bound of the volume, we have

ϵ^{2} ≤ C k r^{β−ν}.

This gives a contradiction if we choose

ln k > ν ln(2b + 1) + ln λ

and let r → 0. q.e.d.

5. Polynomial Growth L-harmonic Functions

In this section, we will consider elliptic operator of divergence form on a complete Riemannian manifold M given by

Lf = div(A(x)∇f)

where A(x) : T_{x}M → T_{x}M is an endomorphism of the tangent space satisfying

⟨A(x)V, V ⟩ > 0

for all nonzero V ∈ T_{x}M. We also assume that in terms of local coordinates, the coefficients of A(x) = (a_{ij}(x)) are measurable functions. Moreover, there exists a constant Ω > 0, such that,

(5.1)  0 < ϕ(r) I ≤ A(x) ≤ Φ(r) I < ∞

for all x ∈ B_{p}(\frac{3r}{2} ) \ B_{p}(\frac{r}{2}), with

(5.2)  \frac{Φ(r)}{ϕ(r)} ≤ Ω.

Theorem 10. Let M be a Riemannian manifold with nonnegative Ricci curvature. Let us define

H_{d}(L, M) = {f | Lf = 0 and |f|(x) = O(r^{d}(x))}

to be the space of L-harmonic functions with polynomial growth of degree at most d. Under the ellipticity assumption (5.1) and (5.2), the space H_{d}(L, M) must be of finite dimension. In fact, there exist constants C_{1}(Ω, n) > 0 and C_{2} > 0 depending only on the described variables such that

dim H_{d}(L, ℝ^{n}) ≤ C_{1} exp(C_{2}d).

Proof. The strategy is to observe that if f is an L-harmonic function, then f^{2} still satisfies the maximum principle. To use the remark preceding Lemma 3, we need to verify that the weak mean value inequality is valid for the operator L. This is indeed the case if one applies the Moser iteration argument on the ball B_{q}(\frac{r}{2}) with q being distance r from the fixed point p ∈ M. In this case, it is known that only the ratio of the upper and lower bounds of the ellipticity constants, Ω, is involved in the dependency of λ. Since we are using the background Riemannian measure, the ratio

\frac{V_{p}(r)}{V_{q}(\frac{r}{2b})} ≤ \frac{V_{q}(2r)}{V_{q}(\frac{r}{2b})}
≤ (4b)^{n}

by the volume comparison theorem. Hence the estimate (1.4) becomes

tr_{A_{\frac{3r}{2}}} A_{r} ≤ λ (4b)^{n}.

Applying Lemma 3, rather its argument, and letting b = \frac{5}{4}, we obtain for all r < R,

det_{A_{R}}A_{r} ≤ (\frac{k r^{β}}{λ 5^{n} R^{β}} )^{k}

if k = dim H_{d}(L, M), with

(5.3)  β = \frac{ln k − ln(λ 5^{n})}{ln(\frac{3}{2})}.

However, the growth assumption of H_{d}(L, M) implies that

det_{A_{r}}A_{R} ≤ CR^{k(2d+n)}.

Hence, together with (5.3), we conclude that

β ≤ 2d + n,

which proves that k must be finite and bounded by

k ≤ C_{1} exp(C_{2} d)

for some constants C_{1}, C_{2} > 0. q.e.d.

We would like to point out that in the previous proof, we have only used the fact that M has a Sobolev inequality and satisfies the volume comparison theorem. These two conditions can replace the assumption that M has nonnegative Ricci curvature. In particular, if M is quasiisometric to a manifold with nonnegative Ricci curvature, then these two properties will hold.

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