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ExamplesAn elementary example of a rigidity theorem is the high school geometry theorem that any two triangles with the same side lengths, must in fact have the same angles as well and are thus congruent; or in non-existence form, one cannot find a pair of triangles with the same side-lengths but differing angles. Thus, triangles are rigid in a way in which quadrilaterals, for instance, are not.

Deeper examples of rigidity theorems include

Liouville type theorems: solutions to a certain PDE which are "uniformly bounded" or "precompact" in some sense are necessarily constant (or trivial, or a soliton...). Such theorems have become a central part of the modern theory of critical dispersive PDE, see e.g. this survey of Killip and Visan. Similar themes also come up in Perelman's proof of the Poincare conjecture, when he finds that the asymptotics of Ricci singularities are in some sense governed by the very special solutions known as gradient shrinking solitons. Somewhat related here are the various elliptic regularity theorems, such as the theorem that every weakly harmonic function (or distribution) is strongly harmonic.
Hilbert's fifth problem: C^0 Lie groups are necessarily C^infty Lie groups (and in particular come with the rich algebraic structure of a Lie algebra, which is certainly not obvious at all if one only begins with C^0 regularity). This is actually part of a large constellation of related theorems, such as Cartan's theorem that any closed subgroup of a Lie group is again a Lie group, or Gromov's theorem that any group of polynomial growth is virtually nilpotent, or the Peter-Weyl theorem which asserts (among other things) that every compact group is the inverse limit of linear Lie groups.
Eliashberg-Gromov symplectic rigidity: roughly speaking, the C^0 limit of symplectomorphisms is again a symplectomorphism, which is surprising as one would naively imagine one would need something like C^2 control instead. I'm not an expert on this topic, but I understand that this is a fundamental theorem in symplectic topology. A related result which also has some rigidity flavour to it (and is certainly a non-existence theorem) is Gromov's symplectic non-squeezing theorem that one cannot sympletically map a large ball into a thin cylinder. (Hmm, Gromov's name is coming up a lot...)
Dynamical rigidity: For certain very special types of dynamics (such as homogeneous dynamics coming from unipotently generated groups), the only minimal closed invariant sets or ergodic measures are those that come from algebraic constructions, such as algebraic subgroups; Ratner's deep theorems on this subject are model examples of this phenomenon. These theorems can be incredibly powerful for establishing equidistribution or density of orbits in homogeneous spaces, for instance the Oppenheim conjecture can be solved as a quick corollary of these theorems (though this was not quite the historical chain of events, as Margulis' proof of this conjecture preceded Ratner's work by a few years). Another example of dynamical rigidity is superrigidity, that in some cases the action of a continuous group can be controlled by the subaction of a discrete lattice; this in turn is related to hyperbolic geometry rigidity theorems such as Mostow rigidity, though this is far from my own area of expertise. Kazhdan's property (T), which asserts that approximately invariant vectors in actions of certain groups must be close to genuinely invariant ones, is another related property of groups that certainly pulls in the direction of rigidity.

Examples of rigidity theorems include

Liouville type theorems: solutions to a certain PDE which are "uniformly bounded" or "precompact" in some sense are necessarily constant (or trivial, or a soliton...). Such theorems have become a central part of the modern theory of critical dispersive PDE, see e.g. this survey of Killip and Visan. Similar themes also come up in Perelman's proof of the Poincare conjecture, when he finds that the asymptotics of Ricci singularities are in some sense governed by the very special solutions known as gradient shrinking solitons. Somewhat related here are the various elliptic regularity theorems, such as the theorem that every weakly harmonic function (or distribution) is strongly harmonic.
Hilbert's fifth problem: C^0 Lie groups are necessarily C^infty Lie groups (and in particular come with the rich algebraic structure of a Lie algebra, which is certainly not obvious at all if one only begins with C^0 regularity). This is actually part of a large constellation of related theorems, such as Cartan's theorem that any closed subgroup of a Lie group is again a Lie group, or Gromov's theorem that any group of polynomial growth is virtually nilpotent.
Eliashberg-Gromov symplectic rigidity: roughly speaking, the C^0 limit of symplectomorphisms is again a symplectomorphism, which is surprising as one would naively imagine one would need something like C^2 control instead. I'm not an expert on this topic, but I understand that this is a fundamental theorem in symplectic topology. A related result which also has some rigidity flavour to it (and is certainly a non-existence theorem) is Gromov's symplectic non-squeezing theorem that one cannot sympletically map a large ball into a thin cylinder. (Hmm, Gromov's name is coming up a lot...)
Dynamical rigidity: For certain very special types of dynamics (such as homogeneous dynamics coming from unipotently generated groups), the only minimal closed invariant sets or ergodic measures are those that come from algebraic constructions, such as algebraic subgroups; Ratner's deep theorems on this subject are model examples of this phenomenon. These theorems can be incredibly powerful for establishing equidistribution or density of orbits in homogeneous spaces, for instance the Oppenheim conjecture can be solved as a quick corollary of these theorems (though this was not quite the historical chain of events, as Margulis' proof of this conjecture preceded Ratner's work by a few years). Another example of dynamical rigidity is superrigidity, that in some cases the action of a continuous group can be controlled by the subaction of a discrete lattice; this in turn is related to hyperbolic geometry rigidity theorems such as Mostow rigidity, though this is far from my own area of expertise. Kazhdan's property (T), which asserts that approximately invariant vectors in actions of certain groups must be close to genuinely invariant ones, is another related property of groups that certainly pulls in the direction of rigidity.

An elementary example of a rigidity theorem is the high school geometry theorem that any two triangles with the same side lengths, must in fact have the same angles as well and are thus congruent; or in non-existence form, one cannot find a pair of triangles with the same side-lengths but differing angles. Thus, triangles are rigid in a way in which quadrilaterals, for instance, are not.

Deeper examples of rigidity theorems include

Liouville type theorems: solutions to a certain PDE which are "uniformly bounded" or "precompact" in some sense are necessarily constant (or trivial, or a soliton...). Such theorems have become a central part of the modern theory of critical dispersive PDE, see e.g. this survey of Killip and Visan. Similar themes also come up in Perelman's proof of the Poincare conjecture, when he finds that the asymptotics of Ricci singularities are in some sense governed by the very special solutions known as gradient shrinking solitons. Somewhat related here are the various elliptic regularity theorems, such as the theorem that every weakly harmonic function (or distribution) is strongly harmonic.
Hilbert's fifth problem: C^0 Lie groups are necessarily C^infty Lie groups (and in particular come with the rich algebraic structure of a Lie algebra, which is certainly not obvious at all if one only begins with C^0 regularity). This is actually part of a large constellation of related theorems, such as Cartan's theorem that any closed subgroup of a Lie group is again a Lie group, or Gromov's theorem that any group of polynomial growth is virtually nilpotent, or the Peter-Weyl theorem which asserts (among other things) that every compact group is the inverse limit of linear Lie groups.
Eliashberg-Gromov symplectic rigidity: roughly speaking, the C^0 limit of symplectomorphisms is again a symplectomorphism, which is surprising as one would naively imagine one would need something like C^2 control instead. I'm not an expert on this topic, but I understand that this is a fundamental theorem in symplectic topology. A related result which also has some rigidity flavour to it (and is certainly a non-existence theorem) is Gromov's symplectic non-squeezing theorem that one cannot sympletically map a large ball into a thin cylinder. (Hmm, Gromov's name is coming up a lot...)
Dynamical rigidity: For certain very special types of dynamics (such as homogeneous dynamics coming from unipotently generated groups), the only minimal closed invariant sets or ergodic measures are those that come from algebraic constructions, such as algebraic subgroups; Ratner's deep theorems on this subject are model examples of this phenomenon. These theorems can be incredibly powerful for establishing equidistribution or density of orbits in homogeneous spaces, for instance the Oppenheim conjecture can be solved as a quick corollary of these theorems (though this was not quite the historical chain of events, as Margulis' proof of this conjecture preceded Ratner's work by a few years). Another example of dynamical rigidity is superrigidity, that in some cases the action of a continuous group can be controlled by the subaction of a discrete lattice; this in turn is related to hyperbolic geometry rigidity theorems such as Mostow rigidity, though this is far from my own area of expertise. Kazhdan's property (T), which asserts that approximately invariant vectors in actions of certain groups must be close to genuinely invariant ones, is another related property of groups that certainly pulls in the direction of rigidity.

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created Jul 6, 2012 at 19:11

Terry Tao

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One important class of "non-existence theorems", which includes Liouville's theorem as a model example, are the various rigidity theorems throughout mathematics that tend to have the general flavour of "any object in class X which is weakly regular, is automatically strongly regular and/or unexpectedly algebraic". Or, in non-existence form, "There does not exist any object of class X which is weakly regular, but not strongly regular". These results tend to be very powerful in applications (because one can then verify a strong regularity property simply by establishing a much weaker, and thus presumably easier to check, regularity property) and are also often important psychological bridges between otherwise disjoint categories (e.g. between the topological and smooth categories). It also often allows one to think of strongly regular objects in X as being "isolated" in some sense from the irregular objects, since there are no transitional objects of intermediate regularity.

Rigidity results can often be motivated as an unexpected converse: one can state the stronger and weaker notions of regularity, and show that the stronger implies the weaker with so much room to spare that it would seem ridiculous that the converse claim can be established - and yet it is true.

Examples of rigidity theorems include

Liouville type theorems: solutions to a certain PDE which are "uniformly bounded" or "precompact" in some sense are necessarily constant (or trivial, or a soliton...). Such theorems have become a central part of the modern theory of critical dispersive PDE, see e.g. this survey of Killip and Visan. Similar themes also come up in Perelman's proof of the Poincare conjecture, when he finds that the asymptotics of Ricci singularities are in some sense governed by the very special solutions known as gradient shrinking solitons. Somewhat related here are the various elliptic regularity theorems, such as the theorem that every weakly harmonic function (or distribution) is strongly harmonic.
Hilbert's fifth problem: C^0 Lie groups are necessarily C^infty Lie groups (and in particular come with the rich algebraic structure of a Lie algebra, which is certainly not obvious at all if one only begins with C^0 regularity). This is actually part of a large constellation of related theorems, such as Cartan's theorem that any closed subgroup of a Lie group is again a Lie group, or Gromov's theorem that any group of polynomial growth is virtually nilpotent.
Eliashberg-Gromov symplectic rigidity: roughly speaking, the C^0 limit of symplectomorphisms is again a symplectomorphism, which is surprising as one would naively imagine one would need something like C^2 control instead. I'm not an expert on this topic, but I understand that this is a fundamental theorem in symplectic topology. A related result which also has some rigidity flavour to it (and is certainly a non-existence theorem) is Gromov's symplectic non-squeezing theorem that one cannot sympletically map a large ball into a thin cylinder. (Hmm, Gromov's name is coming up a lot...)
Dynamical rigidity: For certain very special types of dynamics (such as homogeneous dynamics coming from unipotently generated groups), the only minimal closed invariant sets or ergodic measures are those that come from algebraic constructions, such as algebraic subgroups; Ratner's deep theorems on this subject are model examples of this phenomenon. These theorems can be incredibly powerful for establishing equidistribution or density of orbits in homogeneous spaces, for instance the Oppenheim conjecture can be solved as a quick corollary of these theorems (though this was not quite the historical chain of events, as Margulis' proof of this conjecture preceded Ratner's work by a few years). Another example of dynamical rigidity is superrigidity, that in some cases the action of a continuous group can be controlled by the subaction of a discrete lattice; this in turn is related to hyperbolic geometry rigidity theorems such as Mostow rigidity, though this is far from my own area of expertise. Kazhdan's property (T), which asserts that approximately invariant vectors in actions of certain groups must be close to genuinely invariant ones, is another related property of groups that certainly pulls in the direction of rigidity.