aa r X i v : . [ m a t h . DG ] O c t Heat Kernel for Open Manifolds
Trevor H. JonesNovember 1, 2018
Abstract
In a 1991 paper by Buttig and Eichhorn, the existence and unique-ness of a differential forms heat kernel on open manifolds of boundedgeometry was proven. In that paper, it was shown that the heat kernelobeyed certain properties, one of which was a relationship between thederivative of heat kernel of different degrees. We will give a proof ofthis condition for complete manifolds with Ricci curvature boundedbelow, and then use it to give an integral representation of the heatkernel of degree k . In this paper we are considering the differential forms heat equation onmanifolds, in particular we are considering (∆+ ∂ t ) ω = 0 with Dirchlet initialconditions. Our goal is to produce a formula for the Green’s function, alsoknown as the heat kernel or fundamental solution, which gives the solutionof this equation.The solutions of this equation in the case of functions, or 0-forms, is well-known. The work on differential forms has been much more recent. In 1983,Dodziuk [Dod83] proved that for complete oriented C ∞ Riemannian mani-folds with Ricci curvature bounded below, bounded solutions are uniquelydetermined by their initial values. In 1991 Buttig and Eichhorn [BE91] gaveexistence and uniqueness results for the differential forms heat kernel on openmanifolds of bounded geometry. In the same paper they also gave an identityrelating derivatives of the heat kernel of differing degrees. Using that iden-tity, the author, in his doctoral thesis, [Jon08], was able to give a formulafor the 1-form heat kernel on open Riemann surfaces of bounded geometry.This formula, K ( x , y , t ) = ( I + ∗ x ∗ y ) d x d y Z ∞ t K ( x , y , τ ) dτ , (1)1irectly relates the 1-form heat kernel to the 0-form heat kernel, about whichmore is known.One of the properties given by Buttig and Eichhorn for a global heatkernel was that the heat kernels K k ( x , y , t ) and K k +1 ( x , y , t ) are related by d x K k ( x , y , t ) = d ∗ y K k +1 ( x , y , t ) . (2)In this article, we will present an alternate proof of this property for manifoldswith Ricci curvature bounded below, and then use this to give a formula forthe k -form heat kernel. Lemma 1
For a complete manifold M , with Ricci curvature bounded frombelow, we have the following relationship between the k - and ( k +1) -form heatkernels:1. d x K k ( x , y , t ) = d ∗ y K k +1 ( x , y , t ) d y K k ( x , y , t ) = d ∗ x K k +1 ( x , y , t ) Proof:
Let E ( x , y , t ) = d x K k ( x , y , t ) − d ∗ y K k +1 ( x , y , t ) . We will demonstratethat E satisfies the heat equation with zero as the initial condition. This willimply, by uniqueness of the solutions of the heat equation, see [Dod83], that E ≡
0, giving the desired result.First ∆ x E = ∆ x d x K k ( x , y , t ) − ∆ x d ∗ y K k +1 ( x , y , t )= d x ∆ x K k ( x , y , t ) − d ∗ y ∆ x K k +1 ( x , y , t )= d x ( − ∂ t ) K k ( x , y , t ) − d ∗ y ( − ∂ t ) K k +1 ( x , y , t )= − ∂ t E. Next consider W := h E, ω ( x ) i , where ω is a suitable test function and h µ, ν i = R M µ ∧ ∗ ν . Thenlim t → W = lim t → h d x K k , ω ( x ) i − (cid:10) d ∗ y K k +1 , ω ( x ) (cid:11) = lim t → h K k , d ∗ x ω ( x ) i − d ∗ y h K k +1 , ω ( x ) i = d ∗ y ω ( y ) − d ∗ y ω ( y ) = 0Since ω was an arbitrary test function, we must have that E ≡ t = 0.Thus by uniqueness, E ≡ t > (cid:4) We will use this result to give an explicit formula for K k in terms of K k ± . Theorem 2
Let M be an open, complete manifold with Ricci curvaturebounded below. Then the differential forms heat kernel obey the followingrelation: K k ( x , y , t ) = d x d y Z ∞ t K k − ( x , y , τ ) dτ + d ∗ x d ∗ y Z ∞ t K k +1 ( x , y , τ ) dτ . Proof:
Let K k be the k -form heat kernel. Clearly, K k ( x , y , t ) = − Z ∞ t ∂∂τ K k ( x , y , τ ) dτ , since K k tends to zero (pointwise) as t increases. Since K k is a solution of theheat equation, we can replace the time derivative with − ∆ x = − d x d ∗ x − d ∗ x d x ,so K k ( x , y , t ) = Z ∞ t ( d x d ∗ x + d ∗ x d x ) K k ( x , y , τ ) dτ . Using Lemma 1, we can rewrite the above as K k ( x , y , t ) = Z ∞ t d x d y K k − ( x , y , τ ) + d ∗ x d ∗ y K k +1 ( x , y , τ ) dτ . (cid:4) The result in Theorem 2 depends mainly on two things: the existenceand uniqueness of the heat kernel, and the pointwise convergence to zero ofthe kernel for large time. The methods used above work for a diffusion-typeequation provided these conditions are met. For example, for the diffusionequation (∆ + c∂ t ) ω = 0, the proofs follow through almost identically. Corollary 3
Let M be an open, n -dimensional, differentiable manifold, withRicci curvature bounded below, and consider the differential form diffusionequation (∆ + c∂ t ) ω = 0 with initial data ω ( x ,
0) = f ( x ) . Then the Green’sfunctions are related by G k ( x , y , t ) = d x d y Z ∞ ct G k − ( x , y , τ ) dτ + d ∗ x d ∗ y Z ∞ ct G k +1 ( x , y , τ ) dτ . roof: Let T = ct , then c∂ t = ∂ T , so the equation becomes (∆ + ∂ T ) ω ( x , T ) =0 with the same initial conditions. So by Theorem 2 we have the desiredGreen’s functions. (cid:4) In the case of 2-dimensional manifolds, the 0-form and the 2-form heatkernels are isomorphic, as the following Lemma will show. This allows use towrite the 1-form heat kernel in terms of the 0-form, or function, heat kernel.
Lemma 4
Let M be a complete manifold with Ricci curvature bounded below.Then the differential forms heat kernels, K k and K n − k are related in thefollowing manner: K k = ∗ x ∗ y K n − k . Proof:
Consider the equation ( ∂ t + ∆ k ) u = 0 , u ( x ,
0) = f ( x ). Then u is given by u ( x , t ) = h K k ( x , y , t ) , f ( y ) i = Z M K ( x , y , t ) ∧ y ∗ y f ( y ). Since ∗ ∆ k = ∆ n − k ∗ , it follows that ∗ x u is a solution of the ( n − k )-form heatequation with initial condition ∗ x f ( x ). So ∗ x u ( x , t ) = h K n − k ( x , y , t ) , ∗ y f ( y ) i = h∗ y f ( y ) , K n − k ( x , y , t ) i = Z M ∗ y f ( y ) ∧ y ∗ y K n − k ( x , y , t )By applying ∗ x to both sides, and changing order in the wedge product, wehave ( − k ( n − k ) u ( x , t ) = Z M ( − k ( n − k ) ∗ x ∗ y K n − k ( x , y , t ) ∧ y ∗ y f ( y )or u ( x , t ) = h∗ x ∗ y K n − k ( x , y , t ) , f ( y ) i . By uniqueness of the heat kernel we have the desired result. (cid:4)
Corollary 5
Let M be an open, complete manifold of dimension 2 with Riccicurvature bounded below. Then the 1-form heat kernel on M is given by K ( x , y , t ) = ( I + ∗ x ∗ y ) d x d y Z ∞ t K ( x , y , τ ) dτ where, x , y ∈ M and t > and K is the 0-form heat kernel. roof: Since M has dimension 2, and so K = ∗ x ∗ y K by Lemma 4. Recallthat d ∗ ∗ ω = − ∗ dω for 0-forms, ω . This gives the desired result. (cid:4) As an example, consider the case of the hyperbolic plane, with constantcurvature −
1. From [Cha84] we have the 0-form heat kernel K ( x , y , t ) = 12 π Z ∞ P − + iρ (cosh d H ( x , y )) ρe − ( + ρ ) t tanh πρdρ, which, if we perform the integration set out in Corollary 5, we get K ( x , y , t ) = π ( I + ∗ x ∗ y ) d x d y (cid:20)Z ∞ P −
12 + iρ (cosh d H ( x , y )) ρ e − ( 14 + ρ t
14 + ρ tanh πρdρ (cid:21) . If M is an open 2-dimensional manifold which has a unique heat ker-nel for functions, K , then Corollary 5 suggests a candidate for a heatkernel on 1-forms, and since the K and K heat kernels are isomorphic,we would know all the heat kernels. We will now show that K ( x , y , t ) =( I + ∗ x ∗ y ) d x d y R ∞ t K ( x , y , τ ) dτ works as the heat kernel. So given(∆ (1) x + ∂ t ) w ( x , t ) = 0 (3) w ( x ,
0) = f ( x ) (4)show that w can be written as w ( x , t ) = h K ( x , y , t ) , f ( y ) i .So, let w be a solution of (3) and (4), and w ( x , t ) = h K ( x , y , t ) , f ( y ) i ,Since the Laplacian commutes with the Hodge star isomorphism and theexterior derivative and coderivative, it is clear that w satisfies equation (3).Now we just need to show that w as defined, satisfies the initial condition(4). w ( x , t ) = Z ∞ t d x h d y K ( x , y , τ ) , f ( y ) i + ∗ x d x h∗ y d y K ( x , y , τ ) , f ( y ) i dτ = Z ∞ t d x (cid:10) K ( x , y , τ ) , d ∗ y f ( y ) (cid:11) − ∗ x d x h K ( x , y , τ ) , ∗ y d y f ( y ) i dτ = Z ∞ t d x d ∗ x w ( x , τ ) − ∗ x d x ∗ x d x w ( x , τ ) dτ = Z ∞ t ∆ w ( x , τ ) dτ = Z ∞ t − ∂ τ w ( x , τ ) dτ = w ( x , t )5ince w is a solution of the heat equation with initial value f , and w = w , this means that w also has initial value f . Thus w is a solution of(3) and (4).Finally, let us consider the case of compact complete manifolds. In thiscase, because of conservation, diffusion does not tend to zero, so the large-time limit has to be taken into account. Theorem 6
Let M be a complete manifold with Ricci curvature boundedbelow, and that the lim t →∞ K k ( x , y , t ) is a constant double-form, call it C .Then, the heat kernel obeys the following relation: K k ( x , y , t ) = C + d x d y Z ∞ t K k − ( x , y , τ ) dτ + d ∗ x d ∗ y Z ∞ t K k +1 ( x , y , τ ) dτ . Proof:
Let K k be the k -form heat kernel. Clearly, K k ( x , y , t ) = C − Z ∞ t ∂∂τ K k ( x , y , τ ) dτ , since K k tends to C as t increases. Since K k is a solution of the heat equation,we can replace the time derivative with − ∆ x = − d x d ∗ x − d ∗ x d x , so K k ( x , y , t ) = C + Z ∞ t ( d x d ∗ x + d ∗ x d x ) K k ( x , y , τ ) dτ . Using Lemma 1, we can rewrite the above as K k ( x , y , t ) = C + Z ∞ t d x d y K k − ( x , y , τ ) + d ∗ x d ∗ y K k +1 ( x , y , τ ) dτ . (cid:4) I would like to thank Dr. Kucerovsky for his helpful comments of earlierversions of this paper.
References [BE91] Ingolf Buttig and Jurgen Eichhorn. The heat kernel for p -forms onmanifolds of bounded geometry. Acta Sci. Math. , 55:33–51, 1991.6Cha84] Isaac Chavel.
Eigenvalues in Riemannian Geometry , volume 115 of
Pure and Applied Mathematics . Academic Press, 1984.[Dod83] Jozef Dodziuk. Maximum Principle for Parabolic Inequalities andthe Heat Flow on Open Manifolds.
Indiana University MathematicsJournal , 32(5):703–16, 1983.[Jon08] Trevor H. Jones.