Comparison theorem

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^[1]
• Berger–Kazdan comparison theorem^[2]
• Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)^[3]
• Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures^[4]
• Lichnerowicz comparison theorem
• Eigenvalue comparison theorem
  • Cheng's eigenvalue comparison theorem

See also: Comparison triangle

Other

• 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

References

1. ↑ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
2. ↑ Weisstein, Eric W., "Berger-Kazdan Comparison Theorem", MathWorld.
3. ↑ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
4. ↑ R.L. Bishop & R. Crittenden, Geometry of manifolds