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Trace-free three-dimensional coercivity for the logarithmic gradient

Statement

Let $V$ be a three-dimensional real inner-product space and let $k : V \to V$ be symmetric, meaning $⟨ k x, y ⟩ = ⟨ x, k y ⟩$ for all $x, y \in V$ , and trace free, meaning $tr (k) = 0$ . Write $∣ x ∣^{2} = ⟨ x, x ⟩$ for vectors, $∥ k ∥_{o p} = \max_{∣ x ∣ = 1} ∣ k x ∣$ , and $∣ k ∣^{2} = \sum_{i = 1}^{3} ∣ k e_{i} ∣^{2}$ for any orthonormal basis $(e_{i})_{i = 1}^{3}$ . The last expression, whose basis independence is proved below, defines the Hilbert–Schmidt norm of $k$ .

Then $∥ k ∥_{o p}^{2} \leq \frac{2}{3} ∣ k ∣^{2}$ . If $v \geq 1$ and $K \geq 0$ are real numbers and $Z, Q, A \in V$ satisfy $∣ Z ∣^{2} = v^{2} - 1$ , $∣ Q ∣ \leq K v^{2}$ and $v A = k Z + Q$ , then for every $ε > 0$ one has $∣ A ∣^{2} \leq \frac{2}{3} (1 + ε) ∣ k ∣^{2} + (1 + ε^{- 1}) K^{2} v^{2}$ .

In particular, let $(M, h)$ be a smooth three-dimensional Riemannian manifold, let $v : M \to (0, \infty)$ be smooth, and fix $p \in M$ . Identify $V$ with $(T_{p} M, h_{p})$ , put $A = ({grad}_{h} \log v)_{p}$ , and use $v (p)$ for the numerical $v$ in the preceding hypotheses. Whenever those hypotheses and the vector identity hold at $p$ , the same bound controls $∣ {grad}_{h} \log v ∣_{h}^{2} (p)$ , including when $v (p) = 1$ . The vector identity is an assumption of this application, not a conclusion about arbitrary functions $v$ .

Facts & Assumptions

[F1]

For a linear endomorphism, symmetry is self-adjointness, and a real self-adjoint endomorphism has an orthonormal eigenbasis. Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis

[F2]

The trace of an endomorphism is the basis-independent trace of its matrix; matrix trace is the sum of the diagonal entries. The basis-independent trace of an endomorphism of a finite-dimensional vector space The trace tr(A) as the sum of the diagonal entries

[F3]

Use the inner-product norm for vectors and the maximum over unit vectors for the operator norm. The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖

[F4]

For a smooth function $f$ on a Riemannian manifold, $h_{p} (({grad}_{h} f)_{p}, X) = d f_{p} (X)$ for every $X \in T_{p} M$ . The Riemannian gradient is the metric dual of the differential

[F5]

The real inner product is bilinear and positive definite. Real and complex inner product spaces, with the inner product linear in the first argument

Proof

Given: The symmetric trace-free map, the numerical vector relation, and, for the application, the stated smooth Riemannian data. Goal: Establish both coercivity bounds and their conditional gradient application.

1.1F1F5givenalgebra

For arbitrary $x, y, z \in V$ and $a, b \in R$ , set $d = k (a x + b y) - a k x - b k y$ . The assumed symmetry and [F5] give $⟨ d, z ⟩ = ⟨ a x + b y, k z ⟩ - a ⟨ x, k z ⟩ - b ⟨ y, k z ⟩ = 0$ . Taking $z = d$ and using positive definiteness yields $d = 0$ . Thus $k$ is linear, and its symmetry makes it self-adjoint by [F1]. In particular, its endomorphism trace and operator norm are defined before they are used below.

2.1F1F3step 1.1givenalgebra

By [F1], an orthonormal basis exists. For two such bases write $f_{j} = \sum_{i} o_{i j} e_{i}$ . Expanding each $e_{i}$ in the basis $(f_{j})$ gives $\sum_{j} o_{i j} o_{ℓ j} = δ_{i ℓ}$ , so $\sum_{j} ∣ k f_{j} ∣^{2} = \sum_{i, ℓ, j} o_{i j} o_{ℓ j} ⟨ k e_{i}, k e_{ℓ} ⟩ = \sum_{i} ∣ k e_{i} ∣^{2}$ . This proves that the finite sum defining $∣ k ∣^{2}$ is basis independent.

3.1F1F2F3step 2.1givenalgebra

By [F1], choose an orthonormal eigenbasis with eigenvalues $λ_{1}, λ_{2}, λ_{3}$ , ordered so that $∣ λ_{1} ∣$ is largest. In this basis [F2] and trace freeness give $λ_{2} + λ_{3} = - λ_{1}$ , and step 2.1 gives $∣ k ∣^{2} = λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2}$ . Since $λ_{2}^{2} + λ_{3}^{2} \geq (λ_{2} + λ_{3})^{2} / 2$ , one obtains $∣ k ∣^{2} \geq \frac{3}{2} λ_{1}^{2}$ . For $x = \sum_{i} x_{i} e_{i}$ , $∣ k x ∣^{2} = \sum_{i} λ_{i}^{2} x_{i}^{2} \leq λ_{1}^{2} ∣ x ∣^{2}$ , with equality at $x = e_{1}$ . Thus [F3] gives $∥ k ∥_{o p}^{2} = λ_{1}^{2} \leq \frac{2}{3} ∣ k ∣^{2}$ .

4.1F3step 3.1givenalgebra

For vectors $a, b$ and $ε > 0$ , expanding $∣ \sqrt{ε} a - ε^{- 1 / 2} b ∣^{2} \geq 0$ gives $∣ a + b ∣^{2} \leq (1 + ε) ∣ a ∣^{2} + (1 + ε^{- 1}) ∣ b ∣^{2}$ . Apply this to $A = k Z / v + Q / v$ . The numerical hypotheses directly imply $∣ Z ∣^{2} / v^{2} = 1 - v^{- 2} \leq 1$ and $∣ Q ∣ / v \leq K v$ . By [F3] and step 3.1, $∣ k Z ∣^{2} / v^{2} \leq ∥ k ∥_{o p}^{2} \leq \frac{2}{3} ∣ k ∣^{2}$ . Substitution proves the claimed vector bound. Only division by $v \geq 1$ is used; if $v = 1$ , then $Z = 0$ and the argument still applies.

5.1F4step 4.1given∎

In the stated Riemannian application, positivity and smoothness of $v$ make $\log v$ smooth, and [F4] defines $A = ({grad}_{h} \log v)_{p}$ in the three-dimensional inner-product space $(T_{p} M, h_{p})$ . The assumed pointwise vector identity and bounds therefore satisfy precisely the hypotheses of step 4.1 with the scalar $v (p)$ . That step proves the asserted gradient bound, also at $v (p) = 1$ . No differential identity has been inferred from the numerical assumptions.

Source notes

The Appendix B reference motivates the trace-free method. Its arcosh argument uses $v > 2$ ; the present pointwise algebra divides only by $v$ and proves the conditional bound for every $v \geq 1$ .

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External dependencies

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, proof of Lemma B.1, (B.8)–(B.10), printed/PDF pp.38–39

The proof bounds the square of the largest absolute eigenvalue of the trace-free three-dimensional second fundamental form by its squared norm with coefficient 2/3, then estimates the gradient of arcosh v.

Hypotheses: A spacelike maximal three-dimensional hypersurface, a future unit timelike observer, bounded background coefficients, and v>2 for the arcosh calculation.

Reading limits: Consulted PDF text on pp.38–39 around (B.8)–(B.10); screenshot requests failed. This is methodological background, not an imported all-v theorem. The present proof uses finite-dimensional algebra and an assumed vector identity; it does not differentiate v. Its conditional logarithmic-gradient application and validity at v=1 are established locally. No weighted maximum estimate is imported.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, proof of Lemma B.1, (B.8)–(B.10), printed/PDF pp.38–39