Hodge theory, lecture 6 M. Verbitsky Hodge theory lecture 6: Laplace operator is Fredholm NRU HSE, Moscow Misha Verbitsky, February 10, 2018 1
Hodge theory, lecture 6 M. Verbitsky Fredholm operators (reminder) DEFINITION: A continuous operator F : H 1 − → H 2 of Hilbert spaces is called Fredholm if its image is closed and kernel and cokernel are finite- dimensional. REMARK: “Cokernel” of a morphism F : → H 2 of topological H 1 − H 2 vector spaces is often defined as im F . DEFINITION: An operator F : H 1 − → H 2 has finite rank if its image has finite rank. CLAIM: An operator F : H 1 − → H 2 is Fredholm if and only if there exists F 1 : H 2 − → H 1 such that the operators Id − FF 1 and Id − F 1 F have finite rank. Proof: This is because F defines an isomorphism F : H 1 / ker F − → im F as shown above. 2
Hodge theory, lecture 6 M. Verbitsky Fredholm operators and compact operators (reminder) THEOREM: The set of Fredholm operators is open in the operator norm topology. → V be a Fredholm operator, and U 1 := (ker F ) ⊥ . Proof. Step 1: Let F : U − | F ( x ) | Since F is invertible on U 1 , it satisfies inf x ∈ U 1 > 2 ε . Then, for any | x | | F + A ( x ) | operator A with � A � < ε , one has inf x ∈ U 1 > ε . This implies that | x | � � U 1 is an invertible map to its image, which is closed. In particular, F � ker( F + A ) is finite-dimensional. Step 2: To obtain that coker( F + A ) is finite-dimensional for � A � sufficiently small, we observe that coker( F + A ) = ker( F ∗ + A ∗ ), and F ∗ is also Fred- holm. Then Step 1 implies that ker( F ∗ + A ∗ ) is finite-dimensional for � A � sufficiently small. COROLLARY: Let A be compact and F Fredholm. Then A + F is Fred- holm. Proof: Let A i be a sequence of operators with finite rank converging to A . Then F + ( A − A i ) is Fredholm for i sufficiently big, because the set of Fredholm operators is open. However, a sum of Fredholm operator and operator of finite rank is Fredholm, hence F + A = F + ( A − A i ) + A i is also Fredholm. 3
Hodge theory, lecture 6 M. Verbitsky Equivalent scalar products on vector spaces THEOREM: Let V be a vector space, and g 1 , g 2 two scalar products. We say that g 1 is equivalent to g 2 if these two scalar product induce the same topology. THEOREM: The topology induced by g 1 is equivalent to topology in- duced by g 2 if and only if C − 1 g 2 � g 1 � Cg 2 for some C > 0. Proof: Consider the identity operator A : ( V, g 1 ) − → ( V, g 2 ). Its operator g 2 ( x,x ) norm is sup x � =0 g 1 ( x,x ) . Operator norm is bounded if and only if Id is con- tinuous, and this is equivalent to existence of a constant C > 0 such that C − 1 g 2 � g 1 . Existence of a constant C such that g 1 � Cg 2 is equivalent to continuity of A − 1 . 4
Hodge theory, lecture 6 M. Verbitsky Equivalent scalar products and symmetric operators LEMMA: Let V be a vector space, and g , g 1 scalar products. Consider the symmetric operator B 1 such that g 1 ( x, y ) = g ( B 1 ( x ) , y ). Then � 2 � g ( B 1 ( x ) , B 1 ( x )) g 1 ( x, x ) sup = sup . g ( x, x ) g ( x, x ) x x Proof: By Cauchy-Schwarz, g ( x, x ) g ( B 1 ( x ) , B 1 ( x )) � g ( B 1 ( x ) , x ) 2 = g 1 ( x, x ) 2 . � 2 . On the other hand, sup x g ( B 1 ( x ) ,B 1 ( x )) This gives sup x g ( B 1 ( x ) ,B 1 ( x )) sup x g 1 ( x,x ) � � g ( x,x ) 2 g ( x,x ) g ( x,x ) is norm of B 2 1 , which gives � 2 � g ( B 1 ( x ) , B 1 ( x )) g 1 ( x, x ) 1 � � � B 1 � 2 = sup = � B 2 sup g ( x, x ) g ( x, x ) x x � 2 . hence sup g ( B 1 ( x ) ,B 1 ( x )) sup x g 1 ( x,x ) � � g ( x,x ) 2 g ( x,x ) 5
Hodge theory, lecture 6 M. Verbitsky Equivalent scalar products and Fredholm operators REMARK: A continuous operator F : H 1 − → H 2 in vector spaces with scalar product is called Fredholm if it is Fredholm on their completions (which are Hilbert spaces). Corollary 1: Let g, g 1 , g 2 be metrics on V , and consider the symmetric op- erators B i such that g i ( x, y ) = g ( B i ( x ) , y ). Denote by ˜ g 2 the metric ˜ g 2 ( x, y ) := g 2 ( B 2 ( x ) , B 2 ( y ). Then g 1 is equivalent to g 2 if and only if B 1 : ( V, ˜ g 2 ) − → ( V, g ) is Fredholm. Proof: B 1 : ( V, ˜ g 2 ) − → ( V, g ) is Fredholm if and only if it for some constant C > 0, one has C − 1 g ( B 2 ( x ) , B 2 ( x )) � g ( B 1 ( x ) , B 1 ( x )) � Cg ( B 2 ( x ) , B 2 ( x )). This is the same as C − 1 g 2 ( x, x ) � g 1 ( x, x ) � Cg 2 ( x, x ) by the previous lemma. 6
Hodge theory, lecture 6 M. Verbitsky Sobolev’s L 2 -norm on C ∞ c ( R n ) (reminder) DEFINITION: Denote by C ∞ c ( R n ) the space of smooth functions with com- pact support. For each differential monomial P α = ∂ k 1 ∂ k 2 ... ∂ k n ∂x k 1 ∂x k 2 ∂x k n 1 1 2 consider the corresponding partial derivative P α ( f ) = ∂ k 1 ∂ k 2 ... ∂ k n f. ∂x k 1 ∂x k 2 ∂x k n 1 1 2 Given f ∈ C ∞ c ( R n ), one defines the L 2 p Sobolev’s norm | f | p as follows: � | P α ( f ) | 2 Vol | f | 2 � s = deg P α � p where the sum is taken over all differential monomials P α of degree � p , and Vol = dx 1 ∧ dx 2 ∧ ...dx n - the standard volume form. REMARK: Same formula defines Sobolev’s L 2 -norm L 2 p on the space of smooth functions on a torus T n . 7
Hodge theory, lecture 6 M. Verbitsky Connections (reminder) DEFINITION: Recall that a connection on a bundle B is an operator ∇ : → B ⊗ Λ 1 M satisfying ∇ ( fb ) = b ⊗ d B − f + f ∇ ( b ), where f − → d f is de Rham differential. When X is a vector field, we denote by ∇ X ( b ) ∈ B the term �∇ ( b ) , X � . REMARK: In local coordinates, connection on B is a sum of differential and a form A ∈ End B ⊗ Λ 1 M . Therefore, ∇ X is a derivation along X plus linear endomorphism. This implies that any first order differential operator on B is expressed as a linear combination of the compositions of covariant derivatives ∇ X and linear maps. This follows from the definition of the first order differential operator: by definition, it is a linear combination of partial derivatives combined with linear maps. 8
Hodge theory, lecture 6 M. Verbitsky Connection and a tensor product (reminder) ∇ ∗ → Λ 1 M ⊗ B ∗ on REMARK: A connection ∇ on B gives a connection B ∗ − the dual bundle, by the formula d ( � b, β � ) = �∇ b, β � + � b, ∇ ∗ β � These connections are usually denoted by the same letter ∇ . REMARK: For any tensor bundle B 1 := B ∗ ⊗ B ∗ ⊗ ... ⊗ B ∗ ⊗ B ⊗ B ⊗ ... ⊗ B a connection on B defines a connection on B 1 using the Leibniz formula: ∇ ( b 1 ⊗ b 2 ) = ∇ ( b 1 ) ⊗ b 2 + b 1 ⊗ ∇ ( b 2 ) . 9
Hodge theory, lecture 6 M. Verbitsky L 2 p -metrics and connections DEFINITION: Let F be a vector bundle on a compact manifold. The L 2 p - topology on the space of sections of F is a topology defined by the norm p = � p M |∇ i f | 2 Vol M , for some connection and scalar product | f | p with | f | 2 � i =0 on F and Λ 1 M . REMARK: The metric | f | 2 p is equivalent to the Sobolev’s L 2 p -metric on C ∞ ( M ) . Indeed, all partial derivatives of a function f are expressed through ∇ i f , hence an L 2 -bound on partial derivatives gives L 2 -bound on ∇ i f , and is given by such a bound. From now on, we write ( x, y ) instead of � M ( x, y ) Vol M . This metric is also denoted L 2 ; the space of sections of B with this metric ( B, L 2 ). DEFINITION: We define the Sobolev’s L 2 p -metric on vector bundles by p ( x, y ) = � p L 2 i =0 ( ∇ i ( x ) , ∇ i ( y )). 10
Hodge theory, lecture 6 M. Verbitsky L 2 p -metrics and Fredholm maps First, let’s show that we can drop all terms in this sum, except two. Theorem 1: The Sobolev’s L 2 p -metric is equivalent to g ( x, y ) := ( ∇ p ( x ) , ∇ p ( y )) + ( x, y ) . Step 1: Let D 1 = � p i =0 ∇ i mapping B to �� p i =0 (Λ 1 ) ⊗ p � Proof. ⊗ B and D 2 ( x ) = ∇ p + x mapping B to (Λ 1 M ) ⊗ p ⊗ B ⊕ B . Then L 2 p ( x, y ) = ( D 1 ( x ) , y ) Notice that L 2 p ( x, y ) = ( D ∗ and g ( x, y ) = ( D 2 ( x ) , y ). 1 D 1 x, y ) and g ( x, y ) = ( D ∗ 2 D 2 x, y ). Step 2: To prove that these two metrics are equivalent, we need to show that D ∗ → ( B, L 2 ) is Fredholm, where h ( x, y ) = ( D ∗ 1 D 1 x, D ∗ 2 D 2 : ( B, h ) − 1 D 1 y ) (Corollary 1). Step 3: On a flat torus, the metric h is equivalent to L 2 2 p . Using the same argument as proves the Rellich lemma, we obtain that any differential operator → ( B, L 2 ). Φ of order < 2 p defines a compact operator Φ : ( B, h ) − Step 4: The map D ∗ → ( B, L 2 ) is by definition an isometry, and 1 D 1 : ( B, h ) − D ∗ 1 D 1 − D ∗ 2 D 2 is a differential operator of lower order, which is compact as a → ( B, L 2 ) by the Rellich lemma. Then D ∗ 2 D 2 − D ∗ map ( B, h ) − 1 D 1 is a compact operator, and D ∗ 2 D 2 is Fredholm whenever D ∗ 1 D 1 is Fredholm. 11
Recommend
More recommend
