Comparison theorem

In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.

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. Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign with an inequality sign, form a broad class of such auxiliary relations.^[1][2]

One instance of such theorem was used by Aronson and Weinberger to characterize solutions of Fisher's equation, a reaction-diffusion equation.^[3] Other examples of comparison theorems include:

• Chaplygin's theorem
• Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations
• Lyapunov comparison theorem
• Sturm comparison theorem
• Hille-Wintner comparison theorem

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. ^[4]

• 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^[5]
• Berger–Kazdan comparison theorem^[6]
• Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)^[7]
• Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures^[8]
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
  • Cheng's eigenvalue comparison theorem
• Comparison triangle

See also

• Limit comparison theorem, about convergence of series
• Comparison theorem for integrals, about convergence of integrals
• Zeeman's comparison theorem, a technical tool from the theory of spectral sequences