Some analytic and geometric aspects of the p -Laplacian on Riemannian - - PowerPoint PPT Presentation

Some analytic and geometric aspects of the p-Laplacian on Riemannian manifolds

Stefano Pigola Università dell’Insubria Convegno Nazionale di Analisi Armonica Bardonecchia, 15-19 Giugno 2009

• Prob. 1 Detect the topology of a given Riemannian manifold N from the study
• f the space of smooth maps M ! N. Here M is a suitably chosen manifold.
• Prob. 2 Suppose we are given a domain M and a target N. Characterize

homotopically trivial maps among smooth maps M ! N. Recall: two continuous maps f; g : M ! N are homotopic if there is a continuous deformation H : M [0; 1] ! N of H (x; 0) = f (x) into H (x; 1) = g (x) : Special case: g (x) const: = ) f is homotopically trivial. Strategy: Take f : M ! N. Construct a somewhat “canonical” representa- tive u : M ! N in the homotopy class of f. Then u satis…es some system of

• PDEs. Make some analysis on u and derive information on the original f.

A further natural important question arises

• Prob. 3 How many “canonical” representatives u are there in the homotopy

class of f? This last problem will be considered later.

Notation

M, N are Riemannian manifolds, without boundary, either compact or complete. MRiem, MRic are the Riemann and the Ricci (covariant) tensors of M: MRiem

()

q means MRiemx

• ei; ej; ei; ej

()

q (x), 8x 2 M, and for every orthonormal ei; ej 2 TxM. MRic

()

q means MRicx (v; v)

()

q (x), 8x 2 M, and 8v 2 TxM, with jvj = 1.

Canonical representatives and topology via PDEs

• Ex. 1 (1-dimensional domains) Given a compact N, let f : S1 ! N be a

smooth loop. Suppose f is not homotopically trivial. Hilbert = ) 9 : S1 ! N :

Z

S1 j _

j2 = min

Z

S1 j _

j2 : homotopic to f

• :

The loop satis…es the equation := D _ = 0 i.e. is a smooth closed geodesic. Suppose that there are no energy min- imizer geodesic loops (a kind of Liouville type result). Then N has trivial fundamental group 1 (N) = 0. For instance, we have

• Th. 1 (Synge) N compact, orientable, NRiem > 0, dim N = 2m =

) “Liouville” for energy minimizers= ) 1 (N) = 0.

• Ex. 2 (compact m-dim. domains) Let N be a compact target, NRiem
• 0. Let f : M ! N be a smooth map. If M is compact then

Hartman Eells-Sampson =

) 9u : M ! N :

Z

M jduj2 = min

Z

M jdhj2 : h homotopic to f

• :

The minimizer u satis…es the (system of) equations u := div (du) = 0 i.e. u is a harmonic map. Note: u is smooth by elliptic regularity. Suppose, for some choice of M, the validity of a Liouville type result u = 0 = ) u = const: Then f is trivial from the topological viewpoint. For instance, we have the following

• Th. 2 (Eells-Sampson) M compact, MRic 0 and N compact, NRiem
• 0. If MRicp0 > 0 for some p0 2 M =

) Liouville for harmonic maps = ) every smooth f : M ! N is homotopically trivial.

An application: Cartan theorem. N cmpt, NRiem 0. Every element of 1 (N) has in…nite order. Proof (geometric rigidity of groups). Set Zk := Z=kZ and, by contra- diction, suppose 9Zk 1 (N), for some k 2. Consider the lens space M = S3=Zk. Then MRiem > 0 and 1 (M) ' Zk. Fix any injective ho- momorphism : 1 (M) ' Zk ! 1 (N). Since NRiem 0, by the general theory of Eilenberg-McLane spaces = ) there exists a smooth map f : M ! N which induces , say = f#. Applying the Theorem = ) f homotopically trivial = ) f# = 0 = ) is the trivial homomorphism. Contradiction.

• Rem. 1 Similar arguments can be used to prove Preissman thm and its gen-

eralizations such as Lawson-Yau ‡at torus theorems and so on...

• Ex. 3 (Noncompact domains) Let N be compact, NRiem 0. Let f :

M ! N be a smooth map. Suppose M complete, non-compact(!). Prescribe asymptotic behaviour at in…nity, say jd fj 2 L2 (M) (…nite energy). Schoen-Yau = ) 9u : M ! N :

8 > < > :

u = 0 jduj 2 L2 u homotopic to f: Suppose, for some choice of M, the validity of a Liouville type result u = 0 = ) u = const: Then f is trivial from the topological viewpoint. Liouville for MRic 0 was proved by Schoen-Yau. But an amount of negative curvature is allowed

• Th. 3 ([P.-Rigoli-Setti, J.F.A. 2005]) M complete, non-compact, MRic

q (x) with q(x)

1

(M) 0 and N compact, NRiem 0 = ) Liouville for …nite energy harmonic maps = ) every …nite energy smooth f : M ! N is homotopically trivial.

Proof (Sketch). Starting point: Bochner formula+re…ned Kato (RHS) (*) jduj jduj + q (x) jduj2 1 m jr jdujj2 : By assumption q(x)

1

(M) = inf

'2C1

c (M)

R

jr'j2q(x)'2

R

'2

0: (**) FischerColbrie-Schoen = ) 9v > 0 : v + q (x) v = 0: In the spirit of the generalized maximum principle: w = jduj v = ) div

• v2rw
• 0, 0 w 2 L2

M; v2dvol

• :

L2-Liouville for di¤usion operators= ) w = const:

();()

= ) jduj = const: and

MRic 0 =

) jduj = 0.

What about higher energies?

Natural candidates as “canonical” representatives are p-harmonic maps.

• Def. 1 The p-Laplacian (or p-tension …eld) of a manifold valued u : M ! N,

(possibly N = R) is de…ned by pu = div

• jdujp2 du
• ;

where du 2 1 TM; u1TN

• is a 1-form with values in u1TN, jduj is

the Hilbert-Shmidt norm of du, and div is the formal adjoint of the exterior di¤erential d with respect to the L2-inner product on 1.

• Def. 2 A map u : M ! N is p-harmonic if pu = 0: In case N = R, u is

p-subharmonic if pu 0 and p-superharmonic if pu 0.

• Th. 4 (W.S. Wei) Assume that (M; h; iM) is complete and that (N; h; iN)

is compact with NRiem 0. Fix a smooth map f : M ! N with …nite p-energy jd fjp 2 L1 (M), p 2. Then, in the homotopy class of f, there exists a p-harmonic map u : M ! N, u 2 C1;, with jdujp 2 L1 (M). Therefore, to prove that f is homot.-trivial we use the following very general

• Th. 5 ([P.-Veronelli, Geom. Ded. 2009]) Let (M; h; iM) be complete man-

ifold such that MRic q (x). Let (N; h; i) be a complete(!) manifold with

NRiem 0: Let u : M ! N be a p(> 2)-harmonic map, u 2 C1, such that

Z

BR

jduj

p = o (R) ; R ! +1;

for some p p. If Hq(x)

1

(M) 0; for some H > p2=4 ( p 1), then u is constant.

About the proof. Again we start with a Bochner-type inequality jduj jduj + q (x) jduj2 hdu; dui ; where, since u is p-harmonic, u = (p 2) du (r log jduj) : However: (a) The RHS is not so nice as in the case p = 2 (no sign, no re…ned Kato). The previous technique cannot be applied. We need manipulations in integral form and a direct use of the spectral assumption with suitably chosen test-functions. (b) u is not smooth. We use of a version of the approximation procedure by Duzaar-Fuchs. Idea: C1-approximate u on M+ = fjduj > 0g by smooth uk (not p-harmonic). Prove an L

p-Caccioppoli type inequality fo uk. The

Caccioppoli contains an extra term that vanishes as k ! +1. Take limits to get a Caccioppoli for u. Duzaar-Fuchs teach us how to extend this inequality from M+ to M.

Uniqueness of the “canonical” representative

Good targets for uniqueness are complete manifolds N with NRiem 0: Let f : M ! N be a smooth map. Let u : M ! N be a p-harmonic representative in the homotopy class of f. Assume u (M) nondegenerate, i.e., u (M) 6 a geodesic of N. Then, u is unique in the following situations. Linear case p = 2. (a) (Hartman) M; N compact manifolds, NRiem < 0. (b) (Schoen-Yau) M; N complete manifolds, vol (M) < +1, NRiem < 0, and jduj2 2 L1 (M). Nonlinear case p > 1. (a) (S.W. Wei) M; N compact manifolds, NRiem < 0:

• Rem. 2 In the nonlinear situation p 6= 2, the case of complete, non-compact

manifolds M; N is far from being understood. As a …rst attempt in this direction we consider uniqueness in the homotopy class of a constant map. We have the following result in the spirit of Schoen- Yau’s.

• Th. 6 ([P.-Rigoli-Setti, Math. Z. 2008]) Let (M; h; iM) and (N; h; iN) be

complete Riemannian manifolds. Assume that M satis…es the volume growth condition vol(@Br)1 = 2 L

1 p1 (+1) ;

for some p 2, and that N has NRiem 0. If u : (M; h; iM) ! (N; h; iN) is a p-harmonic map homotopic to a constant and with energy density jdujp 2 L1 (M), then u is a constant map.

Schoen-Yau argument, p = 2

Let u : M ! N be a harmonic map (freely) homotopic to a costant map, say c u (x0). To simplify the exposition, assume N simply connected, hence Cartan-Hadamard. Otherwise, lift u to a 1-equivariant harmonic map

e

u : f M ! f N between universal coverings. Consider (x) = distN (u (x) ; c) : M ! R0. Since N is Cartan-Hadamard then (x) is a convex function. Harmonic maps pull-back convex functions to subharmonic functions. Therefore 0. Consider w =

q

1 + 2: Then, w 0. Moreover, jruj2 2 L1 = ) jrwj2 2 L1 (M). Now use a Liouville-type theorem to deduce w const. This implies const., but (x0) = 0, so that u u (x0) :

• Rem. 3 The same proof works for 2 (u; v) with u; v : M ! N homotopic,

harmonic maps, v 6 const.. We then obtain (u; v) const.

p-harmonic case If p 6= 2, Schoen-Yau argument cannot be applied because of the following

• Th. 7 ([Veronelli, preprint 2009]) There exist Riemannian manifolds M, N,

a convex function H : N ! R and a p-harmonic map u : M ! N, for some p > 2, such that H u : M ! R is not p-subharmonic. Both the manifolds and the (smooth) p-harmonic map are rotationally symmet-

• ric. The proof shows that, for every increasing, rotationally symmetric convex

function H, it holds lim infx!1 p (H u) (x) < 0. To overcome the di¢culty we shall pass from function to a vector …eld related to u. This inspires to previous work by [S. Kawai, Geom. Ded. 1999].

Non-linear potential theory and Lq-vector …elds

• Def. 3 (M; h; i) is said to be p-parabolic if for any real-valued function u 2

W 1;p

loc \ C0,

(

pu 0 supM u < +1 = ) u const: Geometric conditions insuring p-parabolicity rely on volume growth assump-

• tions. Let vol(Br) and vol(@Br) denote the volume and the area of the geo-

desic ball and sphere of redius r , respectively. Then (special case of [Rigoli- Setti, Revista Mat. Iberoam. 2001]) r vol(Br) = 2 L

1 p1 (+1) =

)

6( =

1 vol(@Br) = 2 L

1 p1 (+1) =

)

6( = p-parabolicity.

• Th. 8 (V. Gol’dshtein and M. Troyanov) (M; h; i) is p-parabolic if and only

the following holds. Let X be a vector …eld on M such that: (a) jXj 2 L

p p1 (M), (b) 0

R

M divX +1

(c) divX 2 L1

loc (M) and min (divX; 0) = (divX) 2 L1 (M)

Then

R

M divX = 0:

• Cor. 1 Let (M; h; i) be a p-parabolic manifold. If u satis…es pu 0 and

jrujp 2 L1 (M), then u const:

• Rem. 4 Making use of (the p = 2 version of) the above corollary one can

considerably improve the …nite volume condition, vol (M) < +1; in Schoen- Yau uniqueness result.

As a matter of fact, the above corollary is a very special case of the next analytic result.

• Th. 9 ([P.-Veronelli, preprint 2009]) Suppose that, for some p > 2, M is

p-parabolic. Then, for u; v : M ! R, the following comparison principle holds:

(

pu pv jrujp ; jrvjp 2 L1 (M) = ) u v const:

• Rem. 5 In case u; v : M ! Rn one can prove a similar comparison, namely

(

pu = pv jdujp ; jdvjp 2 L1 (M) = ) u v const: Note that, since Rn is contractible, every map is homotopic to a constant. Therefore, in case pu = 0 = pv, the vector-valued comparison follows from

• ur uniqueness Theorem. The proof of the general vector-valued comparison is
• btained using a re…ned version of the next arguments.

• Proof. The following inequality can be derived from some work by Lindqvist

and resambles Mikljukov inequality for the mean curvature operator:

D

jrujp2 ru jrvjp2 rv; ru rv

E

C jru rvjp : Now, take : R ! [0; 1] such that 0 0 and consider the vector …eld X = (u v)

• jrujp2 ru jrvjp2 rv
• :

(a) X 2 L

p p1 (M)

(b) div X C0 (u v) jru rvjp 0. Gol’dshtein-Troyanov criterion = ) div X = 0 = ) 0 (u v) jru rvjp = 0. Set A = u (x0) v (x0) and choose such that (t) = t on t 2 [A"; A+"]: Then r (u v) = 0 on the connected component of the set fA " < u v < A + "g s.t. x0 2 . Whence, it easily follows u v A

• n = M.

Outline of the proof of the uniqueness result

Let u : M ! N be a p-harmonic map with jdujp 2 L1 (M). Assume that u is homotopic to a constant map u (x0) = y0 2 N. Fix e x0 2 f M over x0 and

f

y0 2 f N over y0. Let e u :

f

M ! f N be the lifting of u such that e u (f x0) = f y0. Then, e u is p-harmonic. Moreover, u homotopic to a constant implies the equivariant property

e

u ( (e x)) = e u (e x) , 8 2 1 (M; x0) : Let k 2 C2 ([0; +1)) be such that k0 0, k00 0 and k (t) =

(

At2 + B if 0 t << 1 t if t 1;

for suitable constants A; B > 0. Let r (e y) = distN (e y; f y0) and de…ne h (e y) = k (r (e y)) :

f

N Cartan-Hadamard= ) h convex on f N and strictly convex near f y0. By the equivariant property of e u it is well de…ned the vector …eld on M Xx = jde

xe

ujp2 f

Mre x (h e

u) ; x 2 M where e x is any point in the …ber over x. It can be veri…ed that (a) jXj 2 L

p p1,

(b) divM (X) 0: and, furthermore, (c) divM (X) (x) = 0 = ) jdu (x)jp2 trf

M

n

Hess e

N (h) (de xe

u; de

xe

u)

• = 0.

Now vol(@Br)1 = 2 L

1 p1 (+1) =

) p-parabolicity K.N.R. crit. = )

(a)+(b)

divM (X) 0: By (c) jdu (x)j = 0 on

n

x 2 M : Hess e

N (h) (e

u (e x)) > 0

• :

Since h is strictly convex near f y0, then u must be constant in a neighborhood

• f x0. Thus u is locally constant, hence a constant map.