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A Heinz mean value estimate from small scalar energy

Statement

Assume Countable Choice. Let $e$ be a nonnegative $C^{2}$ real function on a neighborhood of the closed disk ${\overline{B}}_{r} (0) \subset R^{2}$ , where $r > 0$ . Let $a, b \geq 0$ and suppose $Δ e \geq - a e^{2} - b e, b r^{2} \leq 1, a \int_{B_{r} (0)} e < \frac{3 π}{64} .$ Then $e (0) \leq \frac{16}{3 π r^{2}} \int_{B_{r} (0)} e .$ The small-energy condition is automatic when $a = 0$ . Translation gives the identical assertion for a disk centered at any point. This is a scalar differential-inequality estimate; identifying its hypotheses for a holomorphic-map energy density is a separate step.

Statement references

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: Countable Choice, the nonnegative function $e$ , disk, constants, and inequalities in the statement. Write $E = \int_{B_{r} (0)} e$ .

[F1]

A $C^{2}$ function with nonnegative Laplacian satisfies the ball mean-value inequality, by Classical subharmonic mean value inequalities.

Proof

1.1givenconstructalgebra

If $e (0) = 0$ , the result follows from $E \geq 0$ . Otherwise the continuous function $(r - ∣ z ∣)^{2} e (z)$ attains a positive maximum at some point $z_{0}$ with $∣ z_{0} ∣ < r$ , since it vanishes on the boundary. Set $M = e (z_{0}) > 0$ and $d = (r - ∣ z_{0} ∣) / 2 > 0$ . For $∣ z - z_{0} ∣ \leq d$ , the inequalities $r - ∣ z ∣ \geq d$ and maximality give $e (z) \leq 4 M$ . At the original center maximality also gives $r^{2} e (0) \leq 4 d^{2} M$ . The closed disk ${\overline{B}}_{d} (z_{0})$ lies in the original open disk.

2.1F1step 1.1algebra

On this smaller disk $Δ e \geq - 16 a M^{2} - 4 b M$ . Put $K = 16 a M^{2} + 4 b M$ . The function $e (z) + K ∣ z - z_{0} ∣^{2} / 4$ has nonnegative Laplacian. Applying [F1] to a radius $0 < ρ \leq d$ and using the disk average of $∣ z - z_{0} ∣^{2}$ , which is $ρ^{2} / 2$ , gives $M \leq \frac{1}{π ρ^{2}} \int_{B_{ρ} (z_{0})} e + \frac{K ρ^{2}}{8} = \frac{1}{π ρ^{2}} \int_{B_{ρ} (z_{0})} e + M (2 a M + b / 2) ρ^{2} .$ Write $q = 8 a M + 2 b$ , so the last term is $M q ρ^{2} / 4$ .

3.1step 1.1step 2.1casesalgebra

If $q d^{2} \leq 1$ , including $q = 0$ , choose $ρ = d$ in step 2.1. It gives $3 M / 4 \leq E / (π d^{2})$ , hence $M d^{2} \leq 4 E / (3 π)$ . The center comparison from step 1.1 then proves $e (0) \leq 16 E / (3 π r^{2})$ .

4.1step 1.1step 2.1step 3.1givencasesalgebra∎

It remains to exclude $q d^{2} > 1$ . Since $d \leq r / 2$ and $b r^{2} \leq 1$ , we have $b d^{2} \leq 1 / 4$ . Thus $8 a M d^{2} > 1 - 2 b d^{2} \geq 1 / 2$ , so $a > 0$ and $M > 1 / (16 a d^{2})$ . Consequently $b / M \leq 16 a b d^{2} \leq 4 a$ , and $q / M = 8 a + 2 b / M \leq 16 a$ . Choose $ρ = q^{- 1 / 2} < d$ in step 2.1. It yields $E \geq \int_{B_{ρ} (z_{0})} e \geq \frac{3 π M}{4 q} \geq \frac{3 π}{64 a},$ contradicting the assumed strict inequality $a E < 3 π / 64$ . Therefore the case of step 3.1 always holds. The zero-center case in step 1.1 and the same calculation after translation prove every asserted clause.

Remarks

The earlier bibliography named Dusa McDuff and Dietmar Salamon, J-holomorphic Curves and Symplectic Topology, §4.3, mean value inequality (https://bookstore.ams.org/coll-52). That cited book edition was not available in this review; no reading of it is asserted. The current reference records the actually read geometric context, while the complete scalar/local differential proof above supplies the stated assertion.

Depends on

Dependency tree · 2 levels

Sources

Urs Frauenfelder and Kai Zehmisch, Gromov compactness for holomorphic discs with totally real boundary conditions, v2