Ngô Quốc Anh

October 12, 2013

Bochner-type formula for the conformal Killing operator on manifolds with boundary

Filed under: Uncategorized — Tags: , — Ngô Quốc Anh @ 8:39

Given a Riemannian manifold (M,g) without boundary, in the previous note, we derived a Bochner-type formula for the conformal Killing operator \mathbb L. Precisely, we obtained

\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g= \int_M |\nabla X|^2 dv_g + \left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g

or equivalenlty,

\displaystyle \frac{1}{2}\int_M |\mathbb LX |^2 dv_g=- \int_M {({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} - {\rm Ric}(X,X)d{v_g}}+\left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g.

Today, we try to derive a similar formula for the operator \mathbb L assuming the manifold has boundary \partial M.  Our starting point again is the Bochner formula for vector fields mentioned here, i.e.

\displaystyle\frac{1}{2}\Delta (|X|^2) = |\nabla X|^2 + {\rm div}({\mathbb L_X}g)(X) - {\nabla _X}{\rm div}X - {\rm Ric}(X,X).

Using this and the formula for \Delta (|X|^2) that we derived here, we arrive at

\displaystyle\frac{1}{2}{\Delta _g}(|X{|^2}) = ({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} + |\nabla X{|^2}

which now yields

\displaystyle -{\rm div}({\mathbb L_X}g)(X)=-({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h}- {\nabla _X}{\rm div}X - {\rm Ric}(X,X).

We now integrate both sides over M. First, using our previous calculation, there holds

\displaystyle \int_M {{{\left\langle {X,\nabla (\text{div} X)} \right\rangle }_g}d{v_g}} = - \int_M {|\text{div} X|_g^2d{v_g}} + \int_{\partial M} {\text{div} X{{\left\langle {X,\nu } \right\rangle }_g}d{\sigma _g}}.

Second, as before, we know that

\begin{array}{lcl}\displaystyle -\int_M {\rm div}({\mathbb L_X}g)(X) dv_g &=&\displaystyle -\int_M X_j{\rm div}({\mathbb L_X}g)(\frac{\partial}{\partial x_j}) dv_g \\ &=& \displaystyle -\int_M X_j \nabla_{\frac{\partial}{\partial x_i}}({\mathbb L_X}g)(\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}) dv_g\\ &=& \displaystyle \int_M (\nabla_{\frac{\partial}{\partial x_i}} X_j) ({\mathbb L_X}g)(\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}) dv_g -\int_{\partial M} X_j (\mathbb L_X g)(\nu, \frac{\partial}{\partial x_j}) d\sigma_g\\ &=& \displaystyle\frac{1}{2}\int_M (\nabla_{\frac{\partial}{\partial x_i}} X_j + \nabla_{\frac{\partial}{\partial x_j}} X_i) \mathbb L_X g_{ij} -\int_{\partial M} (\mathbb L_X g)(\nu ,X) d\sigma_g\\ &=& \displaystyle\frac{1}{2}\int_M |\mathbb L_X g|^2 dv_g -\int_{\partial M} (\mathbb L_X g)(\nu ,X) d\sigma_g\end{array}.

Hence, we have proved that

\begin{array}{lcl}\displaystyle\frac{1}{2}\int_M |\mathbb L_X g|^2 dv_g=&-&\displaystyle\int_M ({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} dv_g+\int_M {|\text{div} X|^2d{v_g}}- \int_M {\rm Ric}(X,X) dv_g\\ &+&\displaystyle \int_{\partial M} (\mathbb L_X g)(\nu ,X) d\sigma_g - \int_{\partial M} {\text{div} X{{\left\langle {X,\nu } \right\rangle }_g}d{\sigma _g}}.\end{array}

Finally, making use of the identity

\displaystyle |\mathbb{L}X|^2=|{\mathbb{L}_X}g{|^2} - \frac{4}{n}{( \text{div}X)^2}

and the definition of \mathbb L, that is to say

\displaystyle \mathbb LX(\nu, X) = (\mathbb L_Xg)(\nu, X) - \frac{2}{n} {\rm div} X \langle \nu, X \rangle,

we eventually arrive at

\begin{array}{lcl}\displaystyle\frac{1}{2}\int_M |\mathbb LX |^2 dv_g=&-&\displaystyle\int_M (\Delta_g X^h){X_h} dv_g+ \left(1-\frac{2}{n}\right) \int_M {|\text{div} X|^2d{v_g}}- \int_M {\rm Ric}(X,X) dv_g\\ &+&\displaystyle \int_{\partial M} (\mathbb LX)(\nu ,X) d\sigma_g - \left( 1 - \frac{2}{n} \right) \int_{\partial M} {\text{div} X{{\left\langle {\nu ,X} \right\rangle } }d{\sigma _g}}.\end{array}

This is the so-called the Bochner-type formula for the conformal Killing operator on manifolds with boundary.

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