Isometric immersions and beyond

László Székelyhidi Jr.Universität Leipzig

Fifth Abel ConferenceCelebrating the Mathematical Impact of

John F. Nash Jr. and Louis Nirenberg

Given a smooth Riemannian manifold (Mn, g)

do there exist isometric immersions in some Euclidean space ?RN

isometric: preserving the length of curves.

A problem in geometry

A problem in PDE

do there exist maps such that, in local coordinates, u : Mn ! RN

@iu · @ju = gij i, j = 1 . . . n ?

Given a smooth Riemannian manifold (Mn, g)

PDE aspects / difficulties

• first order system of equations with unknowns • does not naturally fit into any of the usual (elliptic/parabolic/hyperbolic) types • uniqueness?? • local solvability:

N

Schläfli conjecture (1873): solvability for C1

Janet - Cartan Theorem (1927): solvability for

sn := 12n(n+ 1)

N = snC!

N = sn

General conjecture open. For n=2 partial results by Lin (1986), Nakamura (1987), Han-Hong-Lin (2003), Khuri (2007), ….

Prominent (geometric) special case: the Weyl problem

(S2, g) ,! R3

with positive Gauss curvature g > 0

[H. Weyl 1916]

i.e. a convex compact surface

PhD Thesis of Louis Nirenberg (1949)

“The Determination of a Closed Convex Surface Having Given Line Elements”

“The Weyl and Minkowski problems in differential geometry in the large”L. Nirenberg CPAM (1953)

Complete (PDE) solution of the Weyl problem:

Independently solved also by Pogorelov by purely geometric methods…..

The Weyl Problem

The continuity method (Weyl):

(a) Construct a homotopy with positive curvature{gt}t2[0,1]

(b) “openness”: perturbation problem (IFT) for

(c) “closedness”: a priori estimates for second derivatives using convexity

@iu · @ju = gij

(S2, g) ,! R3

Theorem (Nirenberg 1953)

Let be a metric with on . Then there

exists an isometric immersion of .

g 2 C4 g > 0 S2

u 2 C4 (S2, g) ,! R3

(elliptic Monge-Ampére)

improved later by E. Heinz in 1962 to C3

The Weyl Problem

The continuity method (Weyl):

(a) Construct a homotopy with positive curvature{gt}t2[0,1]

(b) “openness”: perturbation problem (IFT) for

(c) “closedness”: a priori estimates for second derivatives using convexity

@iu · @ju = gij

(S2, g) ,! R3

Simpler problem: suppose is a sequence of smooth maps with {uk}k

(@iuk · @juk � gij) ! 0

uk ! u uniformly(i) @iuk · @juk = gij(ii)

Does it follow that ? @iu · @ju = gij

What if (ii) is replaced by (ii’) uniformly ?

The Weyl Problem

The continuity method (Weyl):

(a) Construct a homotopy with positive curvature{gt}t2[0,1]

(b) “openness”: perturbation problem (IFT) for

(c) “closedness”: a priori estimates for second derivatives using convexity

Linearization:

Crucial fact: homogeneous equation has only trivial solutions, i.e.

@iu · @jv + @ju · @iv = 0 8i, j ) v = a⇥ u+ b

@iu · @jv + @ju · @iv = hij

@iu · @ju = gij

“infinitesimal rigidity”

(S2, g) ,! R3

Rigidity for the Weyl Problem

Linearization:

infinitesimal rigidity

(S2, g) ,! R3

Theorem (Cohn-Vossen 1927, Herglotz 1943)

A isometric immersion of with positive Gauss curvature is unique up to rigid motion.

C2 (S2, g)

rigidity @iu · @ju = gij

Theorem (Blaschke 1916)

The linearization around a isometric immersion of with positive Gauss curvature admits no nontrivial solutions.

(S2, g)C2

@iu · @jv + @ju · @iv = 0

Rigidity for the Weyl Problem (S2, g) ,! R3

Theorem (Cohn-Vossen 1927, Herglotz 1943)

A isometric immersion of with positive Gauss curvature is unique upto rigid motion.

C2 (S2, g)

rigidity @iu · @ju = gij

rigidity below C2

• If u 2 C1and u(S2

) is convex

[Pogorelov 1951]

• If u 2 C1and u(S2

) has bounded extrinsic curvature

(N⇤d� finite measure)

• If u 2 C1,↵ ↵ > 2/3 [Borisov 1950s, Conti-De Lellis-Sz. 2004]

Intrinsic versus extrinsic geometry

�iu · �ju = gij

convergence of regularizations:

@iu` · @ju` � (g`)ij = O�`2↵kuk21,↵

�

consequently, can define parallel transport if↵ > 1/2

u 2 C1,↵, g smooth

and intrinsic curvature = extrinsic curvature, if↵ > 2/3

u` := u ⇤ `

Theorem (J. Nash 1954 - N. Kuiper 1955)

Any short embedding can be uniformly approximated by isometric embeddings.

Mn ,! Rn+1

• an example of Gromov’s h-principle • method of proof: convex integration • embeddings are rigid • Lipschitz “version” of theorem is easyC2

Nash-Kuiper embedding

V. Borrelli - S. Jabrane - F. Lazarus - B. Thibert

short: length of curves gets shortened.

C1

The case - the “Nash spiral”

u(x) = u(x) +

a(x)

�

⇣sin(�⇠ · x)⇣(x) + cos(�⇠ · x)⌘(x)

⌘Spiralling perturbation:

amplitudefrequency

@iu · @j u = @iu · @ju+ a2⇠i⇠j +O� 1�

�

Mn ,! Rn+2

Start with a short embedding with normals u : Mn ,! Rn+2 ⌘, ⇣

The case - the “Nash spiral”

u(x) = u(x) +

a(x)

�

⇣sin(�⇠ · x)⇣(x) + cos(�⇠ · x)⌘(x)

⌘Spiralling perturbation:

amplitudefrequency

@iu · @j u = @iu · @ju+ a2⇠i⇠j +O� 1�

�

Mn ,! Rn+2

Start with a short embedding with normals u : Mn ,! Rn+2 ⌘, ⇣

linear part of perturbation

The Nash scheme: an inner iteration

gij � @iu@ju =n⇤X

k=1

a

2k(x)⇠

ki ⇠

kj

u1 = u, u1, u2, . . . , u = un⇤+1 by

uk+1 = uk +

ak

�k

⇣sin(�k⇠

k · x)⇣k + cos(�k⇠k · x)⌘k

⌘

@iuk+1@juk+1 = @iuk@juk + a2k⇠ki ⇠

kj +O

� 1

�k

�

1 << �1 << �2 << · · · << �n⇤

strictly short map:

Define:

so that

frequencies:

.

The Nash scheme: inner iteration

@iu@j u = gij +O� 1

�1

�

ku� ukC0 . 1

�1

ku� ukC1 .n⇤X

k=1

|ak|

. kg � @u@uk1/2C0

final term:

The Nash scheme: outer iteration

v = u+1X

i=1

⇣ n⇤X

k=1

uik

⌘

The outer iteration:

uik is a spiral at frequency �i,k ! 1

PNAS 2012

Theorem (Borisov 1967-2004, Conti-De Lellis-Sz. ‘09) The Nash-Kuiper theorem remains valid for isometric embeddings with

✓ <1

1 + 2n⇤

Nash-Kuiper embedding beyond C1

C1,✓

gij � @iu@ju =n⇤X

k=1

a

2k(x)⇠

ki ⇠

kj

recall:

(for local embeddings in general) n⇤ = 12n(n+ 1)

Theorem (De Lellis-Inauen-Sz. ‘15) For embedding 2-dimensional discs in the Nash-Kuiper theorem remains valid for .

R3

✓ < 1/5

Gromov's h-principle

“What Nash discovered is not any part of the Riemannian geometry, neither it has much (if anything at all) to do with

the classical PDE.”

M. Gromov 2015

h-principle versus local-to-global principle

The main paradigm of the classical physics is non-existence of non-local interactions…..Non-surprisingly, we expect

observable physical patters, e.g. the positions of moving particles after a given time interval, to be predictable in terms of the local laws and a presence of a particular microscopic

law should be manifested by a specific global behaviour.

Classical physics

M. Gromov, 1999

Classical geometry• notion of length • rigidity of closed surfaces / isometric embeddings • …..

The infinitesimal structure of a medium does not effect the global geometry but only the topological behaviour of the medium.

h-principle

M. Gromov, 1999

h-principle versus local-to-global principle

Theorem (J. Nash 1956)

Any short embedding , can be uniformly approximated by smooth isometric embeddings.

u : (Mn, g) ,! RN N � 12n(3n+ 11)

[Nash 1966] Same with real analytic embeddings

The infinitesimal structure of a medium does not effect the global geometry but only the topological behaviour of the medium. The class of infinitesimal laws subjugated by the h-principle is wide, but it does not include most partial differential equations of physics, with a few exceptions leading to unexpected solutions.

h-principle

• PDE in classical physics: Hadamard’s notion of well-posedness, expect uniqueness • PDE in geometry and topology: large invariance group, expect non-uniqueness

cf. Müller, Stefan; Šverák, Vladimir: Unexpected solutions of first and second order partial differential equations. Proceedings of the ICM (Berlin, 1998)

M. Gromov, 1999

h-principle versus local-to-global principle

Nonlinear Elasticity - Non-convex calculus of variations

Z

⌦W (Du) dx

u : ⌦ ⇢ R3 ! R3

⌦ u(⌦)u

Deformation:

Expect non-uniqueness! (e.g. buckling, microstructures,…)

J.M. Ball - R.D. James

Minimize:

Microstructures

C. Chu - R.D. James

How does the microstructure influence macroscopic elastic properties of a material?

Microstructures and differential inclusions

Z

⌦W (Du) dx ! minReplace (wlog , ) W � 0

by

Du(x) 2 K a.e. x

K = {W = 0}

Kframe indifference: is -invariant, i.e. SO(3)

SO(3)K = K

Differential inclusions

K = SO(n)K = SO(2)A [ SO(2)B

K =3[

i=1

SO(3)Ai

Liouville, Reshetnyak ‘68 Friesecke-James-Müller ‘04

Müller-Šverák ‘99

Dolzmann-Kirchheim ‘02

cubic tetragonalcrystal structure:

Du(x) 2 K a.e. x 2 ⌦

“rigidity” “liquid-like behaviour”

Solid-solid phase transitions and the appearance of microstructure:

Differential inclusions

K = SO(n)K = SO(2)A [ SO(2)B

K =3[

i=1

SO(3)Ai

“rigidity” “liquid-like behaviour”

Liouville, Reshetnyak ‘68 Friesecke-James-Müller ‘04

Müller-Šverák ‘99

Dolzmann-Kirchheim ‘02

Solid-solid phase transitions and the appearance of microstructure:

cubic tetragonalcrystal structure:

Du 2 SO(n) a.e. =) u a�ne

8F with |F � Id| < �& detF = 1, 9u :

• Du 2S3

i=1 SO(3)Ai a.e.x 2 ⌦,

• u(x) = Fx on @⌦

Du(x) 2 K a.e. x 2 ⌦

typical statement:

Differential Inclusions

Differential Inclusions

Toy problem: constructv : [0, 1] ! R such that |v| = 1

Baire-category approach(Cellina, Bressan, Dacorogna-Marcellini, Kirchheim)

�v : |v| = 1 a.e.

�v : |v| 1 a.e.

residual in L1 w*

c.f. Mazur Lemma

Differential Inclusions: Baire category approach

Theorem (folklore)

Any 1-Lipschitz map can beuniformly approximated by (weak) Lipschitz isometries, i.e. solutions of

u : ⌦ ⇢ Rn ! Rn

Du(x) 2 O(n) a.e. x

But note: such maps do not necessarily preserve the length of curves!

Theorem (Scheffer 93, Shnirelman 97, De Lellis - Sz. 2009) There exist nontrivial weak solutions of the Euler equations with compact support in space-time.

Non-uniqueness for the Euler equations

@tv + v ·rv +rp = 0

r · v = 0

Classical fact: conservation of energy!1

2

Z|v(x, t)|2 dx

Theorem (Scheffer 93, Shnirelman 97, De Lellis - Sz. 2009) There exist nontrivial weak solutions of the Euler equations with compact support in space-time.

Non-uniqueness for the Euler equations

Theorem (De Lellis - Sz. 2010) Given , there exist infinitely many weak solutions of the Euler equations with

e = e(x, t) > 0

12 |v(x, t)|

2 = e(x, t)

Classical fact: conservation of energy!1

2

Z|v(x, t)|2 dx

Theorem (Wiedemann 2011) For any initial data there exist infinitely many global weak solutions with bounded energy.

L2

• domain is a torus • first global existence result for weak solutions in dimension n � 3

n � 2

Non-uniqueness for the Euler equations

Theorem Given and , there exist infinitely many weak solutions of the Euler equations with

v0 v1

v(t = 0) = v0, v(t = 1) = v1

Selection criteria and instabilities

Theorem (P.L.Lions 1996) Given an initial data , if there exists a solution to the IVP with

then this solution is unique in the class of admissible weak solutions.

v0

⇥v +⇥vT � L�

• The Scheffer-Shnirelman solution clearly not admissible• For the solutions of Wiedemann has an instantaneous jump up at E(t) t = 0

Admissibility:Z

|v(x, t)|2 dx Z

|v0(x)|2 dx

Weak-strong uniqueness

Theorem (De Lellis-Sz. 2010 / Wiedemann-Sz. 2012) There exists a dense set of initial data for which there exist infinitely many admissible weak solutions.

• solutions also satisfy the strong and local energy inequality • a posteriori such initial data needs to be irregular

Wild initial data

v0 2 L2

Theorem (Sz. 2011) There exist infinitely many admissible weak solutions on with initial data given by above. v0

T2

T2

v0(x) =

c.f. Delort (1991): there exists a weak solution with curl v �M+(T2)

Kelvin-Helmholtz instability

Strong instabilities: Kelvin-Helmholtz

Strong instabilities: Kelvin-Helmholtz

turbulent zone

t = 0 t > 0

2t

|v| = 1 a.e.selection principle?

• The solution above is conservative. Strictly dissipative solutions also possible. • Maximal dissipation rate • Maximally dissipative solution is different from vanishing viscosity limit • More realistic limit should include perturbations of the initial condition, i.e. • Key point: the admissibility leads to global constraints for the mean velocity.

(NS⌫) ! (E)

(NS⌫,") ! (E)

dEdt = � 1

6

Strong instabilities: Kelvin-Helmholtz

turbulent zone

t = 0 t > 0

2t

|v| = 1 a.e.

Analogous statements for • compressible isentropic Euler (E. Chiodaroli,C. De Lellis, O. Kreml, E. Feireisl)• Muskat problem / incompressible porous media

(F. Gancedo, D. Faraco, D. Cordoba, Sz.)

Regularity and (turbulent) spectra

Dissipation anomaly in turbulent flows

@tv + div(v ⌦ v) +rp = ⌫�v

div v = 0

• Energy dissipation rate:

• Energy dissipation rate in various turbulent flows seems to remain positive as Reynolds number

• Reynolds number:

✏ = ⌫

Z

T3

|rv|2 dx

Re =UL

⌫

Kolmogorov-Obukhov Spectrum

Energy spectrum:

U. Frisch: Turbulence

Onsager’s conjecture 1949

“In actual liquids this subdivision of energy is intercepted by the action of viscosity, which destroys the energy more rapidly the greater the wave number. However, various experiments indicate that the viscosity has a negligible effect on the primary process; hence one may inquire about the laws of dissipation in an ideal fluid.”

“In fact it is possible to show that the velocity field in such ideal turbulence cannot obey any LIPSCHITZ condition of the form

|v(r0 + r)� v(r0)| (const.)r✓

for any order greater than 1/3.”✓

Onsager’s Conjecture 1949

For (weak) solutions of Euler with

a) If energy is conserved.✓ > 1/3

b) If dissipation possible.✓ < 1/3

Onsager 1949, Eyink 1994,Constantin-E-Titi 1993, Robert-Duchon 2000,Cheskidov-Constantin-Friedlander--Shvydkoy 2007, …

Scheffer 1993, Shnirelman 1999De Lellis - Sz. 2012, Isett 2012

Buckmaster-De Lellis-Isett-Sz 2014Buckmaster 2013, ….

|v(x, t)� v(y, t)| C|x� y|✓

Constantin-E-Titi '93

energy balance

commutator estimate

energy conserved if

Z

T3

12 |v"(T )|

2dx�

Z

T3

12 |v"(0)|

2dx =

Z T

0

Z

T3

rv" · (v" ⌦ v" � (v ⌦ v)") dxdt

kv" ⌦ v" � (v ⌦ v)"k0 . "2✓[v]2✓

@tv" + v" ·rv" +rp" = div�v" ⌦ v" � (v ⌦ v)"

�

Z T

0[v(t)]3C✓ dt < 1 for some ✓ > 1/3

Dissipative Hölder solutions A

Theorem A (Buckmaster - De Lellis - Isett - Sz. ’14) For any smooth positive function e(t) and any there exists a weak solution of the Euler equations such that

and|v(x, t)� v(y, t)| C|x� y|✓

Z

T3

|v(x, t)|2 dx = e(t)

• C. De Lellis - Sz. 2012, same for continuous + 1/10- • S. Daneri - Sz. 2015, non-uniqueness for 1/5-Hölder• A. Choffrut 2013, same in 2D

✓ < 1/5

Dissipative Hölder solutions B

• P. Isett 2012, same with 1/5-Hölder • T. Buckmaster 2013, same with 1/3 Hölder a.e. t• P. Isett - V. Vicol 2014, same for 1/9 Hölder for general class of active scalar equations

Theorem B (Buckmaster - De Lellis - Sz ’15) For any there exists a continuous weak solution of the Euler equations with compact support in time such thatZ 1

0[v(t)]C✓ dt < 1

✓ < 1/3

[v(t)]✓

= supx 6=y

|v(x, t)� v(y, t)||x� y|✓

Nash iteration for Euler

The Euler-Reynolds equations

@tvq +r · (vq ⌦ vq) +rpq = �r ·Rq

r · vq = 0

Euler-Reynolds system: q 2 N

vq+1(x, t) = vq(x, t) +W (x, t,�q+1x,�q+1t)} }slow fast

“fluctuation” - analogue of Nash spiral

The Euler-Reynolds equations

@tvq +r · (vq ⌦ vq) +rpq = �r ·Rq

r · vq = 0

Euler-Reynolds system: q 2 N

more precisely:

vq+1(x, t) = vq(x, t) +W

�vq(x, t), Rq(x, t),�q+1x,�q+1t

�}explicit dependence of previous “state”

Conditions on the "fluctuation"

(H1)

(H2)

(H3)

(H4)

periodic with average zero:⇠ 7! W (v,R, ⇠, ⌧)

W = W (v,R, ⇠, ⌧)

hW i =Z

T3

W (v,R, ⇠, ⌧) d⇠ = 0

hW ⌦W i = R

@⌧W + v ·r⇠W +W ·r⇠W +r⇠P = 0

div⇠W = 0

|W | . |R|1/2 |@vW | . |R|1/2 |@RW | . |R|�1/2

in

prescribed average stress

Estimating new Reynolds stress

(I)

(II)

(III)

Rq+1

Rq+1 = �div�1h@tvq+1 + vq+1 ·rvq+1 +rpq+1

i

�div�1hwq+1 ·rvq

i

�div�1hr ·

�wq+1 ⌦ wq+1 �Rq

�+r(pq+1 � pq)

i

= �div�1h@twq+1 + vq ·rwq+1

i

Estimating new Reynolds stress

“stationary phase” + (H4)

(III) = �div�1hw

q+1 ·rv

q

i= div�1

hX

k

a

k

(x, t)ei�q+1k·xi

k(III)k0 .P

k kakk0�q+1

. kRqk1/20 krvqk0�q+1

(H1)

Rq+1

Estimating new Reynolds stress

(H2)

slow

(H3)

“stationary phase” + (H4)

...and similarlyk(II)k0 = O(1

�q+1)

k(I)k0 = O(1

�q+1)

(II) = �div�1hr ·

�W ⌦W �R

q

�+rP

i= div�1

hX

k 6=0

b

k

(x, t)ei�q+1k·xi

Rq+1

Estimating new Reynolds stress

Summarizing:

vq+1 = vq +W (vq, Rq,�q+1x,�q+1t) + corrector

kvq+1 � vqk0 . kRqk1/20

kRq+1k0 . O(1

�q+1)

1)

2)

…leads to convergence in C0

Rq+1

Conditions on the "fluctuation"

(H1)

(H2)

(H3)

(H4)

periodic with average zero:⇠ 7! W (v,R, ⇠, ⌧)

W = W (v,R, ⇠, ⌧)

hW i =Z

T3

W (v,R, ⇠, ⌧) d⇠ = 0

hW ⌦W i = R

@⌧W + v ·r⇠W +W ·r⇠W +r⇠P = 0

div⇠W = 0

|W | . |R|1/2 |@vW | . |R|1/2 |@RW | . |R|�1/2

in

(family of) stationary solutions: Beltrami flows

convection of microstructure

• h-principle for the isometric immersions: abundance of solutions in case of high codimension or low regularity

• Convex integration: iterative construction, where quadratic is leading order

• Non-uniqueness for Euler connected to strong instabilities (e.g. Kelvin-Helmholtz)

• Selection criteria? Vanishing viscosity limit different from maximally dissipating solution. Mean flow corresponds to “weak closure”

Concluding remarks

h-principle for fluid dynamics

M. Gromov, 1999

The infinitesimal structure of a medium does not effect the global geometry but only the topological behaviour of the medium. The class of infinitesimal laws subjugated by the homotopy principle is wide, but it does not include most partial differential equations of physics, with a few exceptions leading to unexpected solutions. In fact, the presence of the h-principle would invalidate the very idea of a physical law as it yields very limited global information effected by the infinitesimal data.

BUT: the h-principle seems to apply to certain physical situations where a statistical description of a phenomenon (microstructures, turbulence) seems best suited. In terms of the analysis, one is then lead to consider solutions with low regularity.

Thank you for your attention