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The Hodge Theorem and the Bochner Technique: a Vanishingly Short Proof

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The Hodge Theorem and the Bochner Technique: a Vanishingly Short Proof THE HODGE THEOREM AND THE BOCHNER TECHNIQUE: A VANISHINGLY SHORT PROOF HADRIAN QUAN Abstract. This paper serves as a relatively self-contained introduction to the Hodge Decomposition theo- rem, and related techniques in Differential geometry. Assuming minimal exposure to analysis and modern geometry, we motivate and prove some results in the classical theory of an Elliptic Partial Differential Op- erator. This includes constructing the W k;p Sobolev spaces, and proving some related norm inequalities. Armed with the Hodge theorem, we will be able to prove several \Vanishing theorems" of geometric interest. The pacing and tone are ideally written to an undergraduate audience. A number of necessary definitions and results in geometry and analysis are stated or proven in the appendix. Contents 1. A review of Differential forms and de Rham Cohomology 1 2. Preliminaries for Hodge Theory 5 3. An Interlude: The Hodge Theorem as a Least Norm Solution 8 4. Weak Solutions and the Hodge Decomposition Theorem 8 5. Solving Poissons Equation: Laplace and Dirac 11 6. Distributions and The Schwartz Space of Test Functions 17 7. Density, Completeness, and Difference Quotients of W k;p(Ω): The Sobolev Spaces 19 8. Inequalities on Sobolev Norms: Nirenberg and Morrey 22 9. Partial Differential Operators, Weak Existence, and Elliptic Regularity 27 10. The Bochner Technique 30 11. Appendix 32 11.1. A Rapid Introduction to Smooth Manifolds 32 11.2. Riemannian Geometry 33 11.3. Measure Theory 33 11.4. Essentials of Functional Analysis 39 11.5. Orthogonal Decomposition in a Hilbert Space 41 12. References 42 1. A review of Differential forms and de Rham Cohomology Consider the fundamental theorem of calculus, which states that for f 2 C1(R) we have Z b df dx = f(b) − f(a): a dx The student familiar with vector calculus will recall its analogs for 1-dimensional curves C, 2-dimensional surfaces S, and 3-dimensional regions V : Z Z Z rφ · ~tdl = φ(p2) − φ(p1) r × F~ · ~ndS = F~ · ~tdl C S @S Z Z r · F~ dV = F~ · ~ndS V @V where C has endpoints p1; p2 and ~t; ~n are outer normal vectors for a given choice of orientation. In every case we have an equality of two integrals; exchanging the integral of a derivative over a region, for the integral over the boundary of a region. We can generalize these statements, but each of them also involved a more subtle Date: May 2015. 1 2 HADRIAN QUAN phenomenon, namely the ability to integrate. To define the integral we shall introduce differential forms, some of the main players in our program. One reason for this need of differential forms is that even integrals over Rn are not invariant under diffeomorphisms (smooth, bijective maps with smooth inverses). Diffeomorphism is the natural form of equivalence for two smooth manifolds. In the context of calculus, such diffeomorphisms are referred to as `changes of coordinates.' Recall from vector calculus, the change of variables formula states that if f : U ⊂ Rn ! Rn is a local diffeomorphism then Z Z h(x)dx1 : : : dxn = h ◦ f(y) jdet(Df)j dy1 : : : dyn; f(U) U where jdet(Df)j denotes the Jacobian of the map f. Heuristically, if we want to integrate over a manifold we need a mathematical object which transforms similarly to the determinant of a linear map, and resembles the Lebesgue measure dx1 : : : dxn (for more on measure, consult the appendix). Recall first that the dual vector space V ∗ to a vector space V is the space of linear functionals θ : V ! R. These are linear maps which assign a real value to every element of V . One way of defining this space is by defining them with respect to the basis for V . Explicitly, if fv1; : : : vng is a basis for V , then we can uniquely define a basis fθ1; : : : ; θng for V ∗ by requiring ( i 1 i = j θ (vj) = δij = 0 i 6= j and elements of V ∗ can be written with respect to this basis. If we have (x1; : : : ; xn) a choice of local coordinates at a point p 2 M, this generates a basis @ @ ;:::; @x1 @xn for the tangent space TpM at p. Then this basis for the vector space TpM uniquely determines a basis fdx1; : : : ; dxng ∗ for the dual vector space Tp M. For n-dimensional M, there are differential forms of degree from 0 up to n. A degree-k differential form (generally referred to as a k-form) can be defined as iteratively follows. The 0-forms are merely functions of n variables, f(x) = f(x1; : : : ; xn). The 1-forms are all expressible as 1 n α = f1dx + ::: + fndx where the fi coefficients are 0-forms. The 2-forms are of the form 1 n β = α1 ^ dx + ::: + αn ^ dx where the ai coefficients are 1-forms. The wedge product \^" is a multiplication operation between two forms which is associative and distributive, and defined uniquely by i j i j j i i i fidx ^ fjdx = (fifj)dx ^ dx = −(fifj)dx ^ dx dx ^ dx = 0 for all i; j and all functions fi; fj. Continuing as above, we can say in general that k-forms are expressions of the type 1 n ! = 1 ^ dx + ::: + n ^ dx where the i are (k − 1)-forms. What are differential forms intuitively? If we examine 1-forms, our general definition shows they are linear combinations of linear functionals, with coefficients in C1(M; R). So at each point p 2 M the form αp(·): TpM ! R defines a linear functional on TpM as follows: 1 n αp(v) = f1(p)dx (ν) + ::: + fn(p)dx (ν) 8ν 2 TpM: So 1-forms act as smoothly varying linear functionals on each tangent space. The space of degree p-differential forms is denoted ΛpM, and forms a vector space. Similarly, degree k differential forms are smoothly varying linear functionals which take elements of TpM × ::: × TpM (k-tuple) as their arguments. As an example, consider the differential 1-form defined on R2 n f0g xdy − ydx ! = : x2 + y2 THE HODGE THEOREM AND THE BOCHNER TECHNIQUE: A VANISHINGLY SHORT PROOF 3 If we fix the point (1; 1) 2 R2 n 0, it simplies to the linear functional 1 ! (·) = (−dx + dy): T 2 ! ; (1;1) 2 (1;1)R R which takes in tangent vectors based at the point (1; 1), and returns a scalar. Given the above discussion we can now generalize our definition of the integral as follows. Since every n 1 n n differential n−form ! on R can be expressed as !p = f(p)dx ^ ::: ^ dx ; we define the integral of f on R as follows Z Z ! = f(x)dx1 : : : dxn; n n R R where the right hand side is the regular Lebesgue integral of f on Rn. If we want the integral to be finite we typically require that f vanish outside a compact set. This is not any major restriction as any smooth function can be modified in such a way as to remain smooth. The reason for such a definition of integral is that by definition, our differential n−form transforms appropriately as to preserve the change of variables formula. To extend this definition to a manifold, we define similarly to the above, and use the change of variables formula. 1 n If an n−form ! on a subset W ⊂ M can be represented in its chart (W; φ) as !p = f(p)dx ^ ::: ^ dx , we define Z Z ! = f ◦ φ(x)jdet(Dφ)jdx1 : : : dxn; W φ(W ) where dx1 : : : dxn is the n−dimensional Lebesgue measure on Rn. This definition makes sense as U ⊂ M, and φ(W ) ⊂ Rn is an integrable subset. It is a small exercise to show that this definition is independent of the choice of chart, however we shall exclude this for the sake of brevity. Let us return to the land of vector calculus, and resurface to take a gulp of air before diving back into the depths of differential forms. We have three linear differential operators familiar to students, namely the gradient r, curl ∇×, and divergence ∇·. The attentive calculus student may recall that we have the following properties for these operators: for all smooth functions f and all vector fields W that (1.1) r × (rf) = 0 r · (r × W ) = 0: We may recast these statements in a more linear algebraic fashion as follows Im(r) ⊆ Ker(∇×) Im(∇×) ⊆ Ker(∇·); the image of the gradient operator is contained in the kernel of the curl operator, and the image of the curl operator is contained in the the kernel of the divergence operator. In physics terminology, if a vector field X : R3 ! R3 is the gradient of a function (i.e. X = rf) then X is called a conservative field. A vector field Y : R3 ! R3 with zero curl is called irrotational. So (1.1) implies that every conservative field is irrotational, since r × X = r × (rf) = 0. This has the interesting physical interpretation that physical fields of force (which tend to be conservative vector fields) such as the gravitational force or electromagnetic force cannot have any localized rotations in their flows. The phenomena we have observed generalize quite nicely in the spaces of forms. For every differential operator we have described from vector calculus, there is an analog defined on the spaces ΛpM. The exterior derivative, d :ΛpM ! Λp+1M is a linear operator which is deeply related to the spaces of differential forms. It may be defined in an inductive fashion, similarly to differential forms. For a differential p-form ! 2 ΛpM, we say the form is closed if d! = 0, i.e.
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