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 of 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. � M Riem , M Ric are the Riemann and the Ricci (covariant) tensors of M: � � ( � ) ( � ) � M Riem � q means M Riem x e i ; e j ; e i ; e j � q ( x ) , 8 x 2 M , and for every orthonormal e i ; e j 2 T x M . ( � ) ( � ) � M Ric � q means M Ric x ( v; v ) � q ( x ) , 8 x 2 M , and 8 v 2 T x M , with j v j = 1 .

Canonical representatives and topology via PDEs Ex. 1 ( 1 -dimensional domains) Given a compact N , let f : S 1 ! N be a smooth loop. Suppose f is not homotopically trivial. �Z � Z ) 9 � : S 1 ! N : � j 2 = min � j 2 : � homotopic to f Hilbert = S 1 j _ S 1 j _ : 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, N Riem > 0 , dim N = 2 m = ) “Liouville” for energy minimizers = ) � 1 ( N ) = 0 .

Ex. 2 (compact m -dim. domains) Let N be a compact target, N Riem � 0 . Let f : M ! N be a smooth map. If M is compact then Z �Z � Hartman M j du j 2 = min M j dh j 2 : h homotopic to f Eells-Sampson = ) 9 u : M ! N : : 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, M Ric � 0 and N compact, N Riem � 0 . If M Ric p 0 > 0 for some p 0 2 M = ) Liouville for harmonic maps = ) every smooth f : M ! N is homotopically trivial.

An application: Cartan theorem. N cmpt, N Riem � 0 . Every element of � 1 ( N ) has in…nite order. Proof (geometric rigidity of groups). S et Z k := Z =k Z and, by contra- diction, suppose 9 Z k � � 1 ( N ) , for some k � 2 . Consider the lens space M = S 3 = Z k . Then M Riem > 0 and � 1 ( M ) ' Z k . Fix any injective ho- momorphism � : � 1 ( M ) ' Z k ! � 1 ( N ) . Since N Riem � 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, N Riem � 0 . Let f : M ! N be a smooth map. Suppose M complete, non-compact(!). Prescribe f j 2 L 2 ( M ) ( …nite energy) . asymptotic behaviour at in…nity, say j d 8 > � u = 0 < j du j 2 L 2 Schoen-Yau = ) 9 u : M ! N : > : 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 M Ric � 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, M Ric � � q ( x ) with � � � � q ( x ) ( M ) � 0 and N compact, N Riem � 0 = ) Liouville 1 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) j du j � j du j + q ( x ) j du j 2 � 1 m jr j du jj 2 : (*) By assumption R jr ' j 2 � q ( x ) ' 2 � � � � q ( x ) R ( M ) = inf � 0 : 1 ' 2 ' 2 C 1 c ( M ) (**) FischerColbrie-Schoen = ) 9 v > 0 : � v + q ( x ) v = 0 : In the spirit of the generalized maximum principle: � � � 0 , 0 � w 2 L 2 � � w = j du j v 2 r w M; v 2 d vol = ) div : v ( � ) ; ( �� ) L 2 -Liouville for di¤usion operators = ) w = const : = j du j = const : and ) M Ric � 0 = ) j du j = 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 � � j du j p � 2 du � p u = div ; where du 2 � 1 � � TM � ; u � 1 TN is a 1 -form with values in u � 1 TN , j du j is the Hilbert-Shmidt norm of du , and � div is the formal adjoint of the exterior di¤erential d with respect to the L 2 -inner product on � 1 . Def. 2 A map u : M ! N is p -harmonic if � p u = 0 : In case N = R , u is p -subharmonic if � p u � 0 and p -superharmonic if � p u � 0 .

Th. 4 (W.S. Wei) Assume that ( M; h ; i M ) is complete and that ( N; h ; i N ) is compact with N Riem � 0 . Fix a smooth map f : M ! N with …nite f j p 2 L 1 ( M ) , p � 2 . Then, in the homotopy class of f , there p -energy j d exists a p -harmonic map u : M ! N , u 2 C 1 ;� , with j du j p 2 L 1 ( 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 ; i M ) be complete man- ifold such that M Ric � � q ( x ) . Let ( N; h ; i ) be a complete(!) manifold with N Riem � 0 : Let u : M ! N be a p ( > 2) -harmonic map, u 2 C 1 , such that Z p = o ( R ) ; R ! + 1 ; j du j � B R for some � p � p . If � � � � Hq ( x ) ( M ) � 0 ; 1 p 2 = 4 (� for some H > � p � 1) , then u is constant.

About the proof. Again we start with a Bochner-type inequality j du j � j du j + q ( x ) j du j 2 � � h du; d � u i ; where, since u is p -harmonic, � u = � ( p � 2) du ( r log j du j ) : 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: C 1 -approximate u on M + = fj du j > 0 g by smooth u k ( not p -harmonic). Prove an L � p -Caccioppoli type inequality fo u k . 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 N Riem � 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, N Riem < 0 . (b) (Schoen-Yau) M; N complete manifolds, vol ( M ) < + 1 , N Riem < 0 , and j du j 2 2 L 1 ( M ) . Nonlinear case p > 1 . (a) (S.W. Wei) M; N compact manifolds, N Riem < 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 ; i M ) and ( N; h ; i N ) be complete Riemannian manifolds. Assume that M satis…es the volume growth condition 1 vol( @B r ) � 1 = p � 1 (+ 1 ) ; 2 L for some p � 2 , and that N has N Riem � 0 . If u : ( M; h ; i M ) ! ( N; h ; i N ) is a p -harmonic map homotopic to a constant and with energy density j du j p 2 L 1 ( 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 ( x 0 ) . To simplify the exposition, assume N simply connected, hence Cartan-Hadamard. Otherwise, lift u to a � 1 -equivariant harmonic map u : f M ! f e N between universal coverings. Consider � ( x ) = dist N ( u ( x ) ; c ) : M ! R � 0 . Since N is Cartan-Hadamard then � ( x ) is a convex function. Harmonic maps pull-back convex functions to subharmonic functions. q 1 + � 2 : Then, � w � 0 . Therefore � � � 0 . Consider w = Moreover, jr u j 2 2 L 1 = ) jr w j 2 2 L 1 ( M ) . Now use a Liouville-type theorem to deduce w � const . This implies � � const ., but � ( x 0 ) = 0 , so that u � u ( x 0 ) : 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 .