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From hyperbolic angle to logarithmic angle inequalities

Statement

On a Riemannian hypersurface let $v > 2$ be smooth, put $η = arcosh v = \log (v + \sqrt{v^{2} - 1})$ , and suppose $Δ η \geq \coth η (\frac{1}{4} ∣ \nabla η ∣^{2} - C v^{2})$ for some $C \geq 0$ . Then $f = \log v$ satisfies $Δ f \geq \frac{1}{4} ∣ \nabla f ∣^{2} - C v^{2}$ . The convention for gradient and Laplacian is that of the temporal angle data.

Facts & Assumptions

[F1]

Use the positive-trace scalar Laplacian and its gradient from Temporal normal angle and compact positive-height portions. The differential inequality in the Statement is an explicit hypothesis of this algebraic lemma, not a geometric conclusion proved here.

[F2]

The hypersurface carries a smooth positive-definite metric $h$ in the general sense of Riemannian metric and riemannian manifold, allowing an actual boundary and smoothness up to it. The metric-dual gradient, the coordinate Hessian formula $({Hess}_{h} u)_{i j} = \partial_{i} \partial_{j} u - Γ^{k}_{i j} \partial_{k} u$ , and the divergence connection formula are supplied by Gradient hessian and divergence connection formulas. The identity $Δ_{h} u = {tr}_{h} {Hess}_{h} u = {div}_{h} {grad}_{h} u$ and its smooth one-sided boundary convention are Laplace–Beltrami operator as the trace of the Hessian. Only the intrinsic operator convention is borrowed from the temporal angle data; no ambient Lorentzian or maximality assumption is used here.

[F3]

The scalar sum, product, quotient and chain rules are Sum, product, quotient, and chain rules from difference quotients, applied to each coordinate derivative.

[F4]

The inverse identity for $arcosh$ on $[1, \infty)$ is math:thm-inverse-hyperbolic-logarithm-formulas. The hyperbolic identities, quotient definitions and derivatives, including $(\cosh s)^{'} = \sinh s$ , $(\tanh s)^{'} = {sech}^{2} s$ and $0 < \tanh s < 1$ for $s > 0$ , are Addition formulas, identities, parity, and derivatives of the hyperbolic functions. The formula $(\log x)^{'} = 1 / x$ for $x > 0$ is The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t; positive-base powers and their derivative formula, used for the square root on $(0, \infty)$ , are Continuity and derivatives of positive-base real powers. Iterating these derivative formulas with [F3] gives smoothness on the indicated open domains.

Proof

Given: The smooth function and differential inequality in the Statement. Goal: Derive the logarithmic inequality.

1.1F3F4givenalgebra

Since $v > 2$ , [F4] makes $η = arcosh v$ smooth and positive, with $v = \cosh η$ . Set $H (s) = \log \cosh s$ , so $f = H (η)$ . The scalar rules and the logarithm and hyperbolic derivatives give $H^{'} (s) = \sinh s / \cosh s = \tanh s$ and $H^{''} (s) = {sech}^{2} s$ .

2.1F1F2F3step 1.1algebra

In an arbitrary interior coordinate chart, the scalar chain and product rules give $\partial_{i} (H (η)) = H^{'} (η) \partial_{i} η$ and $\partial_{i} \partial_{j} (H (η)) = H^{''} (η) \partial_{i} η \partial_{j} η + H^{'} (η) \partial_{i} \partial_{j} η$ . Substitution in the coordinate Hessian formula yields ${Hess}_{h} (H (η)) = H^{''} (η) d η \otimes d η + H^{'} (η) {Hess}_{h} η$ . Taking the metric trace and using the metric-dual characterization of the gradient gives $\nabla f = \tanh η \nabla η$ and $Δ f = \tanh η Δ η + {sech}^{2} η ∣ \nabla η ∣^{2}$ . At any actual boundary point these identities follow by continuity from the interior in a boundary half-chart: the metric, inverse metric, function derivatives and operator coefficients are smooth up to that point, and the operators there have the one-sided convention in [F1] and [F2].

3.1F1F2F4step 2.1givenalgebra∎

Multiply the assumed lower bound for $Δ η$ by $\tanh η > 0$ . Since $\tanh η \coth η = 1$ , step 2.1 gives $Δ f \geq (\frac{1}{4} + {sech}^{2} η) ∣ \nabla η ∣^{2} - C v^{2} \geq \frac{1}{4} ∣ \nabla f ∣^{2} - C v^{2}$ , where the last step uses $∣ \tanh η ∣ \leq 1$ and the nonnegativity of the squared norm. This also holds at actual boundary points under the stated smooth one-sided convention.

Depends on

Used by · 2

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