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Mathematical Autobiography Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

The Masterpieces of John Forbes Nash Jr.

verfasst von : Camillo De Lellis

Erschienen in: The Abel Prize 2013-2017

Verlag: Springer International Publishing

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Abstract

In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader.

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Vorheriges Kapitel Autobiography
Nächstes Kapitel A Few of Louis Nirenberg’s Many Contributions to the Theory of Partial Differential Equations
Fußnoten
1
In a short autobiographical note, cf. [80, Ch. 2], Nash states that he made his important discovery while completing his PhD at Princeton. In his own words “…I was fortunate enough, besides developing the idea which led to “NonCooperative Games”, also to make a nice discovery relating manifolds and real algebraic varieties. So, I was prepared actually for the possibility that the game theory work would not be regarded as acceptable as a thesis in the mathematics department and then that I could realize the objective of a Ph.D. thesis with the other results.”
 
2
Here we are using the nontrivial fact that in a connected real analytic manifold any pair of points can be joined by a real analytic arc. One simple argument goes as follows: use first Whitney’s theorem to assume, without loss of generality, that Σ is a real analytic submanifold of \(\mathbb R^N\). Fix two points p and q and use the existence of a real analytic projection in a neighborhood of Σ to reduce our claim to the existence of a real analytic arc connecting any two points inside a connected open subset of the Euclidean space. Finally use the Weierstrass polynomial approximation theorem to show the latter claim.
 
3
The projection of an algebraic subvariety is not always an algebraic subvariety: here as well we are taking advantage of the genericity of the projection.
 
4
Many thanks to Riccardo Ghiloni for suggesting this argument, which follows closely the proof of [55, Lem. 3.2].
 
5
Observe that in this context the closure in the Euclidean topology coincides with the Zariski closure.
 
6
Closed manifolds can be C 1 isometrically immersed in lower dimension: already at the time of Nash’s paper this could be shown in \(\mathbb R^{2n-1}\) (for n > 1!) using Whitney’s immersion theorem. Nowadays we can use Cohen’s solution of the immersion conjecture to lower the dimension to n − a(n), where a(n) is the number of 1’s in the binary expansion of n, cf. [17].
 
7
This is what Nash calls “a stage”, cf. [73, p. 391].
 
8
Although the term is nowadays rather common, it was not introduced by Nash, neither in [73] nor in the subsequent paper [75].
 
9
In his paper Nash claims indeed a much larger K(n), cf. [73, bottom of p. 386].
 
10
The argument of Nash is slightly different, since it covers the space of positive definite matrices with appropriate simplices.
 
11
Nash cites Steenrod’s classical book, [94].
 
12
Nash writes Also they could be obtained by orthogonal propagation, cf. [73, top of p. 387].
 
13
The term free was not coined by Nash, but introduced later in the literature by Gromov.
 
14
It must be observed that Nash employs this fact without explicitly stating it and he does not prove it neither he gives a reference. He uses it twice, once in the proof of Theorem 29 and once in the proof of Proposition 35, and although in the first case one could appeal to a more elementary argument, I could not see an easier way in the second.
 
15
Indeed Nash does not give any argument and just refers to a similar reasoning that he uses in Proposition 35 below.
 
16
Nash suggests an alternative argument which avoids the discussion of the dimensions of \(\mathcal {C} (p, L)\) and \(\mathcal {B}\). One can apply his result on real algebraic varieties to find an embedding v which realizes v(Σ) as a real algebraic submanifold, cf. Theorem 1. Then any set of coefficients \(A^r_{ij}\) which is algebraically independent over the minimal field \(\mathbb F\) of definition of v(Σ) (see Proposition 12) belongs to the complement of \(\mathcal {B}\). Since \(\mathbb F\) is finitely generated over the rationals (see Proposition 12), it has countable cardinality and the conclusion follows easily.
 
17
In Nash’s paper the operator is called S θ, where θ corresponds to ε −1. Since it is nowadays rather unusual to parametrize a family of convolutions as Nash does, I have switched to a more modern convention.
 
18
Nash does not take advantage of this simple remark and introduces instead a rather unusual notation to keep track of all the estimates for the intermediate norms in the bounds corresponding to (59), (60) and (61).
 
19
In fact, De Giorgi’s statement is stronger, since in his theorem ∥v∥∞ in (125) is replaced by the L 2 norm of v (note that the power of r should be suitably adjusted: the reader can easily guess the correct exponent using the invariance of the statement under the transformation u r(x) = u(rx)).
 
20
Indeed, it was known that the first partial derivatives of the minimizer satisfy a uniformly elliptic partial differential equation with measurable coefficients. De Giorgi’s stronger version of Theorem 50 would then directly imply the desired Hölder estimate. Nash’s version was also sufficient, because a theorem of Stampacchia guaranteed the local boundedness of the first partial derivatives, cf. [93].
 
21
Nash does not provide any argument nor reference, he only briefly mentions that Theorem 48 follows from Theorem 51 using a regularization scheme and the maximum principle. Note that a derivation of the latter under the weak regularity assumptions of Theorem 48 is, however, not entirely trivial: in Sect. 5.8 we give an alternative argument based on a suitable energy estimate.
 
22
In order to simplify the notation we omit the domain of integration when it is the entire space.
 
23
The first two equations are the first two equations from [77, p. 487, (1)] whereas the third should correspond to [77, p. 488, (1c)]. The latter is derived by Nash from the third equation in [77, p. 487, (1)], which in turn corresponds to the classical conservation law for the entropy, see, for instance, [61, (49.5)]. The third equation of [77, p. 487, (1)] contains two typos, which disappear in [77, p. 488, (1c)]. The latter however contains another error: Nash has η and ζ in place of \(\frac {\eta }{\rho T S_T}\) and \(\frac {\zeta }{\rho T S_T}\), but it is easy to see that this would not be consistent with the way he describes its derivation.
Nash’s error has no real consequence for the rest of the note, since he treats the coefficients in front of \(\mathcal {S} (v)_{ij} \mathcal {S} (v)_{ij}\) and (div v)2 as arbitrary real analytic functions of ρ and T and the same holds for \(\frac {\eta }{\rho T S_T}\) and \(\frac {\zeta }{\rho T S_T}\) under the assumption S T ≠ 0. The latter inequality is needed in any case even to treat Nash’s “wrong” equation for T.
 
24
Indeed Nash does not mention the positivity of S T, although this is certainly required by his argument when he reduces the existence of solutions of (240) to the existence of a solutions of a suitable parabolic system, cf. [77, (6) and (7)]: the equation in T is parabolic if and only if \(\frac {\varkappa }{\rho T S_T}\) is positive.
I also have the impression that his argument does not really need the positivity of S and p, although these are quite natural assumptions from the thermodynamical point of view.
 
25
In the modern literature it is customary to take an equivalent definition of X through formal power series; we refer to [58] for the latter and for several important subtleties related to variants of the Nash arc space.
 
26
In fact, Nash claims the proposition with any algebraic subset W of V in place of V s but, although the proposition does hold for W = V s, it turns out to be false for a general algebraic subset W; cf. [21, Ex. 3.7] for a simple explicit counterexample.
 
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Metadaten
Titel
The Masterpieces of John Forbes Nash Jr.
verfasst von
Camillo De Lellis
Copyright-Jahr
2019
Buch
The Abel Prize 2013-2017

Print ISBN: 978-3-319-99027-9

Electronic ISBN: 978-3-319-99028-6

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-319-99028-6_19