In mathematics, **comparison theorems** are theorems whose statement involves comparisons between various mathematical objects of the same type,^{ [1] } and often occur in fields such as calculus, differential equations and Riemannian geometry.

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.^{ [2] }^{ [3] }

- Chaplygin inequality
^{ [4] } - 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.
- Hille-Wintner comparison theorem

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. ^{ [5] }

- 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
^{ [6] } - Berger–Kazdan comparison theorem
^{ [7] } - Warner comparison theorem for lengths of N-Jacobi fields (
*N*being a submanifold of a complete Riemannian manifold)^{ [8] } - Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures
^{ [9] } - Lichnerowicz comparison theorem
- Eigenvalue comparison theorem
- See also: Comparison triangle

- 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

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

In mathematics, the **uniformization theorem** says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group **Z**^{2}; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.

In the mathematical field of differential geometry, the **Ricci flow**, sometimes also referred to as **Hamilton's Ricci flow**, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation; however, it exhibits many phenomena not present in the study of the heat equation. Many results for Ricci flow have also been shown for the mean curvature flow of hypersurfaces.

**Shing-Tung Yau** is an American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.

**Richard Streit Hamilton** is Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. He made foundational contributions to the theory of the Ricci flow and its use in the resolution of the Poincaré conjecture and geometrization conjecture in the field of geometric topology.

**Mikhael Leonidovich Gromov** is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University.

**Shiu-Yuen Cheng** (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from University of California at Berkeley. Cheng then spent some years as a post-doctoral fellow and assistant professor at Princeton University and the State University of New York at Stony Brook. Then he became a full professor at University of California at Los Angeles. Cheng chaired the Mathematics departments of both the Chinese University of Hong Kong and the Hong Kong University of Science and Technology in the 1990s. In 2004, he became the Dean of Science at HKUST. In 2012, he became a fellow of the American Mathematical Society.

**Geometric analysis** is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.

In Riemannian geometry, a branch of mathematics, **harmonic coordinates** are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties.

In mathematics, the **Berger–Kazdan comparison theorem** is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the *m*-dimensional sphere with its usual "round" metric. The theorem is named after the mathematicians Marcel Berger and Jerry Kazdan.

**Jerry Lawrence Kazdan** is an American mathematician noted for his work in differential geometry and the study of partial differential equations. His contributions include the Berger–Kazdan comparison theorem, which was a key step in the proof of the Blaschke conjecture and the classification of Wiedersehen manifolds. His best-known work, done in collaboration with Frank Warner, dealt with the problem of prescribing the scalar curvature of a Riemannian metric.

**Robert "Bob" Osserman** was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces.

**Spectral geometry** is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.

In mathematics, the **intrinsic flat distance** is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral currents lying in Euclidean space.

**Gerhard Huisken** is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.

**David Allen Hoffman** is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University. In 1985, together with William Meeks, he proved that Costa's surface was embedded. He is a fellow of the American Mathematical Society since 2018, for "contributions to differential geometry, particularly minimal surface theory, and for pioneering the use of computer graphics as an aid to research." He was awarded the Chauvenet Prize in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces". He obtained his Ph.D. from Stanford University in 1971 under the supervision of Robert Osserman.

**Joel Spruck** is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971.

**Frank Wilson Warner** III is an American mathematician, specializing in differential geometry.

- ↑ "The Definitive Glossary of Higher Mathematical Jargon — Theorem".
*Math Vault*. 2019-08-01. Retrieved 2019-12-13. - ↑ "Comparison theorem - Encyclopedia of Mathematics".
*www.encyclopediaofmath.org*. Retrieved 2019-12-13. - ↑ See also: Lyapunov comparison principle
- ↑ "Differential inequality - Encyclopedia of Mathematics".
*www.encyclopediaofmath.org*. Retrieved 2019-12-13. - ↑ Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
- ↑ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
- ↑ Weisstein, Eric W. "Berger-Kazdan Comparison Theorem".
*MathWorld*. - ↑ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
- ↑ R.L. Bishop & R. Crittenden,
*Geometry of manifolds*

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