CMU-CNA-18

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Existence and regularity for solutions to Hyperbolic

Monge-Ampere equations with applications to

Non-Euclidean elasticity.

CMU CNA Seminar September 11, 2018

Shankar Venkataramani

Toby Shearman

Ken Yamamoto

John Gemmer

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My Ph.D students

John Gemmer

Toby Shearman

(3)

φ

h

′2

+

r

′2

= 1

N

= (

h

′

,

−r

′

)

= (sin(

θ

)

,

cos(

θ

))

dω

= sin(

θ

)

dθ

∧

dφ

=

−

d

cos(

θ

)

∧

dφ

=

r

′′

ds

∧

dφ

dA

=

d

φ

∧

ds

⇒

r

′′

=

−

k

(

s

)

r

In addition, we require that

1



r

0



1.

If

k(s)

c >

0, then we cannot have solutions on the entire real line. At

some finite

s

we have

r

0

=

±

1.

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Nonexistence results for isometric immersions of

the Hyperbolic plane

•

Hilbert

(1901): There is no real analytic isometric immersion of

H

2

onto a

complete subset of

R

3

•

Holmgren

(1902): Given a (local) smooth embedding of

H

2

in

R

3

, the

embedding cannot be extended isometrically and smoothly beyond is a

finite distance

d

.

•

Amsler

(1955): Every su

ﬃ

ciently smooth immersion of the hyperbolic plane

into

R

3

has a singular “edge”,

i.e

, a one-dimensional submanifold beyond

which the embedding cannot be smoothly extended.

•

Efimov

(1962): No

C

2

isometric embedding of

H

2

, or any complete

mani-fold with curvature that is uniformly negative in

R

3

.

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Overview

•

Quick review of differential geometry

•

C

1,1

isometric immersions, branch points.

•

Construction of branched surfaces -

Discrete differential

geometry

.

•

Continuum mechanics for discrete surfaces.

•

Applications to geometry, dynamics.

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x

y

u

v

w

Φ

Elastic energy of a thin sheet

E

=

!

∥

γ

∥

2

+

ϵ

2

∥

κ

∥

2

κ

= ˆ

n

·

D

2

Φ

γ

= (

D

Φ)

T

·

D

Φ

−

g

(9)

Elastic energy

E

t

[ ] =

S

[ ] +

t

2

B

[

H, K

]

=

Z

⌦

Q

( )

dxdy

+

t

2

Z

⌦

(4

H

2

K

)

dxdy,

Lewicka and Pakzad (2011).

–limit:

lim

t

!

0

t

2

E

t

=

(R

⌦

(4

H

2

K

)

dxdy

2

W

2

,

2

iso

+

1

otherwise

x

y

u

v

w

Φ

Immersion

:

⌦

!

R

3

of the center surface.

Reference Riemannian metric

g

.

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N

(

p

)

·

d

r

(

p

) = 0

I

≡

ds

2

=

d

r

(

p

)

·

d

r

(

p

)

Geometry: The Gauss Normal map

p

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Hyperbolic Monge Ampere equations

n

= (

w

x

, w

y

,

1)

N

=

k

n

k

.

dA

= (

w

xx

w

yy

w

xy

2

)

dx

^

dy

.

d

=

k

dA

n

k

4

dp

^

dq

=

K

F vK

(

x, y

)

dx

^

dy

.

dp

^

dq

=

K

k

(

n

x,y

k

4

)

dx

^

dy

For any domain on which the normal map is one-to-one, the area of the

spherical image cannot exceed 2⇡

. No such restriction for the planar image.

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Hyperbolic Monge-Ampere equations

Au

xx

+ 2

Bu

xy

+

Cu

yy

+

D

(

u

xx

u

yy

u

2

xy

) +

E

= 0

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Elliptic:

AC

B

2

DE >

0

Hyperbolic:

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AC

B

2

DE <

0

M

=

R

5

with coordinates (

p, q, u, x, y

). A Monge-Ampere equation is an

EDS generate by the contact form

✓

=

du

pdx

qdy

and a 2-form

=

Adp

^

dy

+

B

(

dq

^

dy

dp

^

dx

)

Cdq

^

dx

+

Ddp

^

dq

+

Edx

^

dy,

that is linearly independent from

d

✓

mod

✓

.

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(14)

Hyperbolic surfaces: A quadratic saddle

(15)

Negatively curved sheets: Disk geometry

Small slopes approximation:

det(

⇥⇥

w

) = 1

Solutions:

w

=

1

2

⇣

ax

2

y

a

2

⌘

.

w

= 0 for

y

=

±

ax

. Pick

a

= cot(

/n

).

(16)

Piecewise quadratic surfaces

w

(

x, y

) =

(

xy

y

2

cot(

✓

+

))

0



✓



✓

+

xy

+

y

2

cot(

✓

))

✓



✓



0

Let us consider solutions of det(

D

2

w

) =

1

w

is

C

1

,

1

.

All the straight lines

through any point

(17)

4

see Fig. 4(e-f). Note that, if a hyperbolic surface is C2, every point is locally a (regular) saddle (as in Fig. 4(a)) and there-fore cannot contain branch points. Non-C2 immersions are therefore qualitatively di↵erent from C2 immersions in that

they admit 3-saddles (“monkey saddles”) and higher order saddles, which can mediate a local refinement of the buckling wavelength (See Fig. 5).

FIG. 4. (a-b) Small slope isometric immersions w0

4(x1, x2) and w0₄(x₁, x₂) for constant Gaussian curvature K = 1. w0₄(x₁, x₂) is

con-structed by taking odd periodic reflections of the piece of w0

4(x1, x2)

bounded between the green lines. The mesh on both of these sur-faces correspond to their asymptotic lines. (c-d) Projection of the asymptotic lines of w0

4(x1, x2) and w04(x1, x2) onto the x1, x2 plane.

(e-f) Direction of the gradient rw along circles centered at the ori-gin. The regular saddle in (a) corresponds to a gradient field with winding number -1, so the gradient map is 1 to 1. The 4-saddle in (b) has winding number -3, so the gradient map is a 3 sheeted covering near the origin.

Multiple branch points can be introduced on the surface by replicating the above process at any point, not just the origin. For example, consider the surface w0₂(x1, x2) = x1x2 which is ruled by the asymptotic lines x₁, x₂ = const. A branch point

can be added at (x₁, x₂) = (1/ p2,1/ p2) by removing the

sec-tor x₁, x₂ 1/ p2 and in this region fitting three rotated and

translated copies of w0₆(x₁, x₂) = x₂(x₁ p3x₂) so that the

resulting surface has continuous partial derivatives across the cut; see Fig 5(a). Three more branch points b₂,1, b2,2, b2,3

at a radial distance of 1/4 from b₁_,₁ can be added along rays

emanating from b₁,1 that bisect the lines of inflection; see Fig

5(b). This construction can be continued so that at the n-th it-eration 3n new branch points are added at a radial distance of (1/2)n from the previous branch points. The surface w(x₁, x₂)

formed in the limit n _{! 1} is a fractal with an infinite number of subwrinkles in the region x₁ 0, x₂ 0, x2₁ + x₂2 1, and

it satisfies [w, w] = 1. The solution can be extended by odd

periodic reflections to give a small-slopes isometric immer-sion of the unit disk with K = 1. To illustrate the wrinkling

behavior near the edge we map w to a strip geometry through a conformal map h[x + iy] = w[ex+iy]; see Figs. 5(c-d).

FIG. 5. Finite bending energy solutions to the Monge-Ampere equa-tion [w0_,_w0_] ₌ _{1. (a) Three subwrinkle solution created by}

insert-ing three rotated and translated copies of the solution w0

6(x1, x2) = x₂(x₁ p3x₂) onto the solution w0

2(x1, x2) = x1x2 at a branch point.

(b) Nine subwrinkle solution created by inserting nine copies of

w0

12(x1, x2) = x2(x1 (2 + p

3)x₂) at three branch points added onto the three subwrinkle solution. (c) Extension of the nine subwrinkle solution to the full circular domain. (d) The nine subwrinkle solution mapped to the strip geometry by a conformal map.

The existence of self-similar isometric immersions has im-plications to the modeling of non-Euclidean elastic sheets. As for the strip with = 1, the solution w0₂(x₁, x₂) is

har-monic yet the extension of w0₂(x₁, x₂) to an exact

isomet-ric immersion has divergent bending energy for R ' 1.25

with the bending content concentrated near the singular point x₁ = x₂ ⇡ 1.25/ p2 [22]. We can isometrically immerse disks

with larger R by a global refinement of the wavelength i.e taking n > 2. These solutions increase the bending energy

globally. An energetically favorable alternative might be to introduce a branch point in the n = 2 solution near the

singu-lar point, and locally refining the wavelength instead. Indeed, numerics for = 1/3 in the strip geometry indicate that, even

within the small slopes approximation, localized self similar wrinkling profiles may be energetically preferred over global refinement of the wavelength [2, 23].

Crumpled sheets have an energy scale t5/3 _{which is}

inter-mediate between the stretching and bending energies [32, 33]. In contrast, the existence of W2,2 _{isometric immersions for}

Index of a branch point

C

2

isometries are not

dense in

W

2

,

2

isometries!

Consequences for Numerics?

(18)

Amsler surfaces

Sine-Gordon equation:

uv

= sin

.

Self-similar reduction:

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(

u, v

) = (2

p

uv

)

, z

= 2

p

uv.

Painlev´e III:

zz

+

1

z

= sin

.

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(19)

Hyperbolic disks: Full geometry

By allowing non-smooth embeddings, we can decrease the

curvature, and the energy.

(20)

Geodesics and Asymptotic lines: C

2

surfaces

(21)

Constructing K = -1 surfaces

N

=

N

(

u

,

v

),

N

⋅

N

= 1

∂

u

N

⊥

N

,

∂

v

N

⊥

N

∂

u

r

=

∂

u

N

×

N

,

∂

v

r

=

− ∂

v

N

×

N

Lelieuvre formulae

∂

u

N

× ∂

v

N

=

− ∂

u

r

× ∂

v

r

=

−

sin(

φ

)

|

∂

u

r

||

∂

v

r

|

N

(22)

(23)

“Material coordinates” for

ℍ

2

0 ⇡ 4 ⇡ 2 3⇡ 4 ⇡ 5⇡ 4 3⇡ 2 7⇡ 4

Poincare disk model:

g

=

4

dz d

z

¯

(1

−

|

z

|

2

)

2

Asymptotic coordinates to Material coordinates (u, v

)

7!

z

(u, v).

∂

uv

z

=

z

¯

∂

u

z

∂

v

z

(24)

The art of M.C. Escher

Mapping the Hyperbolic plane

(25)

(26)

Quad-mesh

Although we have used the notation

for a mapping

r

:

Z

2

!

R

3

, this makes

sense as a map

r

:

G

!

R

3

, where

G

⇢

R

2

is a connected planar graph where every vertex has even degree.

Red curve is a Hamburger

Polygon

(27)

(28)

(29)

(30)

(31)

A distributional Sine-Gordon equation

The angle

'

between the asymptotic directions satisfies:

'

uv

= sin

'

jumps across the “lines of inflexion”

!

=

u

du

v

dv

(32)

Hilbert’s Theorem

I

Hilbert (1901): In

E

3

there is no complete, analytic surface

of constant negative curvature.

I

Efimov (1968): In

E

3

there is no complete,

C

2

surface with

curvature

K



c <

0

.

I

Nash (1954): In

E

3

there exist complete,

C

1

surfaces with

K



c <

0

“Theorem” (A Quantitative Hilbert’s Theorem (TS, SV))

For the elastic energy given previously,

i)

inf

C1,1

E

.

exp

R

1/2

ii)

inf

C2

E

&

exp

R

E

=

k

H

2

k

1

(33)

↵

=

⇣

_⇤

⌘

p

+

=

+

↵

+

=

↵

⇣

_⇤

⌘

p

&

⇤ p

!

(34)

An upper bound over

C

1

,

1

surfaces:

A recursion formula

:

n+1

=

n

✓

1 +

↵

4

◆

s

_n+1

=

s

_n

2

↵

2

cos

n

2

with

↵

=

ln

⇤

n

(35)

0 2 4 6 8 10 R, normalized radius

100 101 102 W 2 ,

1 -Energy _C2_-Catenoid

Periodic Amsler ecpR

Branched

0.0 2.5 5.0 7.5 10.0

R

, normalized radius

(36)

(37)

Dynamic geometries

Hydrogel with

programmed geometry.

Eran Sharon et al,

Hebrew University.

(38)

(39)

What does all this mean?

•

Free sheets:

the relevant singularities are

branch points

and

lines of inflection

.

The tangent plane is continuous and the

normal is Lipschitz.

•

Unique —

they

do not concentrate elastic energy

in the

vanishing thickness limit. They are not picked out by the

Gamma-limit.

•

Degeneracy:

The construction with branch points is very

flexible. These sheets have lots of isometries with finite

bending energy, so thin free sheets should be very floppy.

Need a statistical mechanical description.

•

Extrinsic flow:

”Uniformization” of immersions of

(40)

Thank you for your attention!