Comparison theorem

Updated on Oct 01, 2024

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A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.

Differential equations

In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof) provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. See also Lyapunov comparison principle

Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations.

Sturm comparison theorem

Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation.

Riemannian geometry

In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.

Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.

Toponogov's theorem

Myers's theorem

Hessian comparison theorem

Laplacian comparison theorem

Morse–Schoenberg comparison theorem

Berger comparison theorem, Rauch–Berger comparison theorem

Berger–Kazdan comparison theorem

Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)

Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures

Lichnerowicz comparison theorem

Eigenvalue comparison theorem