Some vanishing and …niteness results on complete manifolds: a generalization of the Bochner technique Stefano Pigola Convegno Nazionale di Analisi Armonica, 22-25 maggio 2007
In this talk we present some results, recently obtained in collaboration with M. Rigoli and A.G. Setti, that extend the original Bochner technique to the case of L p harmonic forms on geodesically complete manifolds and in the presence of an amount of negative curvature. Basic references. [1] S. P., M. Rigoli, A.G. Setti, Vanishing theorems on Riemannian manifolds and applications. J. Funct. Anal. 229 (2005), 424–461. [2] S. P., M. Rigoli, A.G. Setti, A …niteness theorem for the space of L p harmonic sections. To appear in Rev. Mat. Iberoamer. [3] S. P., M. Rigoli, A.G. Setti, Topics in geometric analysis: vanishing and …niteness results on complete manifolds. Book in preparation.
We will move in the realm of Geometric Analysis . Roughly speaking: you are given a geometric problem. Summarize it into a family of functions (of geometric content) which, in turn, are governed by a system of di¤erential (in)equalities. Obtain information on the qualitative and quantitative properties of solutions of these di¤erential systems. Geometry, in general, will impose some further constrains and guide the analysis of solutions. Apply this information to the given geometric functions and get a conclusion about the original problem. A prototypical example: the celebrated Bochner technique, originally intro- duced by S. Bochner in the ’50s to investigate the relation between the topology and the curvature of a closed (i.e. compact and without boundary) Riemannian manifold.
The case of a closed manifold Bochner original argument Original question: may one prescribe the sign of the curvature on a generic smooth, closed manifold? Let M be a smooth, compact manifold. Then, there is a contractible, open set E � M , with E = M , such that E supports a metric with constant curvature of a prescribed sign. Simply …x any metric ( ; ) on M , a reference origin p 2 M and delete from M the corresponding cut-locus cut ( p ) , which is a closed (hence compact) set of zero-measure. Thus E = M � cut ( p ) is di¤eomorphic to the star-shaped, relatively compact, open set 0 2 E � T p M � R m via the exponential map exp p . To conclude, …x a constant curvature metric on E and pull it back on E .
Remark. In a (quite strong) sense, the topology of M is contained in the (apparently evanescent) removed set cut ( p ) , e.g., the inclusion i : cut ( p ) , ! M induces isomorphisms between homology (and cohomology) groups H k ( cut ( p ) ; Z ) ' H k ( M ; Z ) at least for k 6 = m; m � 1 . Now, closing M � cut ( p ) by addition of cut ( p ) produces a non-trivial topology that, in general, may represent an obstruction for M to support a Riemannian metric with some curvature bound, e.g., given sign. Bochner result goes pre- cisely in this direction. Let us recall the argument.
Theorem 1 (Bochner) Let ( M; h ; i ) be a connected, closed, oriented, Rie- mannian manifold, m = dim M . Set b 1 ( M ; R ) for the …rst (real) Betti num- ber of M: Then ) b 1 ( M ; R ) � m Ric � 0 on M = the equality holding if and only if M is a ‡at torus. Furthermore, ) b 1 ( M ; R ) = 0 : Ric > 0 at some p 2 M =
Proof. (From Geometry to Analysis) De…ne the Hodge-Laplacian as � H ! = ( d� + �d ) ! = 0 : where d is the exterior di¤erential and � stands for the (formal) adjoint of d with respect to the L 2 inner product of k -forms. Set H k ( M ) = f k -forms ! : � H ! = 0 g ; the vector space of harmonic k -forms on M . By Hodge-de Rham theory b 1 ( M ; R ) = dim H 1 ( M ) : Weitzenbock-Bochner formula states that, for ! 2 H 1 ( M ) , � ! # ; ! # � 1 2� j ! j 2 = j D! j 2 + Ric (BW) ;
where � is the Laplace-Beltrami operator ( + d 2 =dx 2 on R ) and D denotes the extension to 1 -forms of the Levi-Civita connection of M . Suppose Ric � 0 . By assumptions and (BW) � j ! j 2 � 0 ; i.e,. j ! j 2 subharmonic Note that: M closed implies j ! j = const. Two di¤erent viewpoints: (a) L 1 viewpoint. The smooth function j ! j attains its maximum at some point and, therefore, by the Hopf maximum principle we conclude that j ! j = const. (b) L p viewpoint. Use the divergence theorem : Z Z Z � r j ! j 2 � � � r j ! j 2 � � � j ! j 2 r j ! j 2 � 2 � 0 : 2 + j ! j 2 � j ! j 2 � � � � � 0 = M div = � � M M This again implies j ! j = const.
Use this information into (BW) formula: ( BW ) = ) D! = 0 = ) ! is determined by its value at any p 2 M . The evaluation map " p ( ! ) = ! p : H 1 ( M ) ! � 1 � � T � Fix p . p M is an injective homomorphism. Therefore dim H 1 ( M ) � m: Note that � � ! # p ; ! # ( BW ) = = 0 ; at p: ) Ric p Therefore, ) dim H 1 ( M ) = 0 : Ric ( p ) > 0 = ) ! p = 0 = ) ! � 0 =
Remark. Crucial fact in the above proof: Ric � 0 = ) � j ! j � 0 . Question. What happens in the presence of an amount of negative curvature? Answer. In general, there is no uniform bound of dim H 1 ( M ) , i.e., no uniform control on the topology. Example. Let S be an orientable, closed Riemann surface of genus g � 2 , by uniformization (and recalling the Gauss-Bonnet theorem) we can endow S with a Riemannian metric of Gauss curvature � 1 .
Analytical counterpart. Set � R ( x ) = min Ric x ( v; v ) ; v 2 S m � 1 � T x M the pointwise lower bound of the Ricci tensor. From (BW) we have the Bochner inequality 1 2� j ! j 2 + R ( x ) j ! j 2 � j D! j 2 � 0 : (*) We would like to get LHS ( � ) = 0 : But, in general, the maximum principle fails to hold for inequalities of this type. Divergence theorem does not help us. Remark. A fundamental result by M. Gromov [Comm. Math. Helv. 1981] states that a uniform limitation on the Betti numbers of a close manifold is obtained by requiring a control on a further Riemannian invariant, namely, the diameter. From a di¤erent (more analytic) perspective, we shall see momen- tarily how one could think of extending Bochner estimating theorem in the presence of (a little amount of) negative curvature.
Generalized maximum principle We are given a solution � 0 of (*) � + q ( x ) � 0 : Main assumption. Assume M supports a function ' > 0 such that � ' + q ( x ) ' � 0 : Remark. The existence of ' is related to spectral properties of the operator � � � q ( x ) . Idea. To absorb the linear term of (*) using a combination of the solutions and ' .
Set u = ' = ) � u + hr u; r log ' i � 0 : There is no linear (i.e. zero-order) term in u . Therefore, the Hopf maximum principle applies. ) u attains maximum Hopf M closed = = ) u � const: Equivalently, = C'; C � 0 : Use the di¤erential inequalities satis…ed by and ' and deduce that, in fact, � + q ( x ) = 0 :
Special case: ! 2 H 1 ( M ) ; = j ! j 2 ; q ( x ) = 2 R ( x ) ; with R ( x ) the lower Ricci bound. Bochner inequality yields 0 = 1 2� j ! j 2 + R ( x ) j ! j 2 � j D! j 2 � 0 : Once again, ! is parallel, thus extending the original Bochner result. Remark. Let M be closed (parabolic su¢ces). If R ( x ) � 0 , then � ' � � 2 R ( x ) ' � 0 : The superharmonic function ' > 0 must be contstant. Hence R ( x ) � 0 . As a consequence (in the compact - parabolic - setting) the Main assuption represents a genuine extension of Bochner condition Ric � 0 only in case R ( x ) changes its sign.
The setting of open manifolds Question. What does of the previous picture survive in the case of a non- compact manifold ( M; h ; i ) ? Examples help us to understand the situation. We shall consider the general case of harmonic k -forms, any k . First, we need to introduce some more notations and inequalities.
Bochner(-type) and Kato inequalities Let ( M; h ; i ) be any manifold, m = dim M . We consider k -forms, any k . Assume: (a) case k = 1 ; Ric � � R ( x ) : (b) case k > 1 , � x � � R ( x ) ; where � x : � 2 ( T x M ) ! � 2 ( T x M ) is the curvature operator. Take ! 2 H k ( M ) . Then, by Gallot-Meyer [J. Math. pures et appl. 1973], the following Bochner inequality holds 1 2� j ! j 2 + CR ( x ) j ! j 2 � j D! j 2 � 0 ; for a suitable C = C ( k; m ) > 0 . E.g. C = k ( m � k ) if M = H m � 1 .
Direct computations show that j ! j f � j ! j + CR ( x ) j ! jg � j D! j 2 � jr j ! jj 2 ; The sign of the RHS: in general, one has the Kato inequality j D! j 2 � jr j ! jj 2 � 0 : In case ! is both closed and co-closed, i.e., d! = 0 ; �! = 0 ; then we have the re…ned Kato inequality j D! j 2 � jr j ! jj 2 � A jr j ! jj 2 ; for a suitable constant A = A ( m; k ) > 0 . E.g. k = 1 = ) A = 1 = ( m � 1)
Notation. n ! 2 H k ( M ) : j ! j 2 L p o L p H k ( M ) = : Remarks. 1. Alexandru-Rugina [Rend. Sem. Mat. Univ. Politec. Torino 1996] ! 2 L p 6 =2 H k ( M ) 6 = ) d! = 0 nor �! = 0 : 2. Ga¤ney [Annals 1954] Global integration by parts: 9 ! 2 L 2 H k ( M ) > = ; = ) d! = 0 ; �! = 0 + > ( M; h ; i ) complete = ) Re…ned Kato.
Conclusion: take ! 2 H k ( M ) . Then = j ! j � 0 satis…es an inequality of the form f � + q ( x ) g � A jr j 2 ; (*) with q ( x ) 2 C 0 , and A 2 R : We shall refer to (*) as the general Bochner-type inequality .
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